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Counting results for sparse pseudorandom hypergraphs II
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Counting results for sparse pseudorandom hypergraphs II Yoshiharu Kohayakawa a , Guilherme Oliveira Mota a , Mathias Schacht b , Anusch Taraz c a b c
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany Institut für Mathematik, Technische Universität Hamburg–Harburg, Hamburg, Germany
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abstract We present a variant of a universality result of Rödl (1986) for sparse, 3-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ‘‘pseudorandom’’ hypergraphs, which is proved in the companion paper (Counting results for sparse pseudorandom hypergraphs I). © 2017 Elsevier Ltd. All rights reserved.
1. Introduction We say that a graph G = (V , E ) satisfies property Q(η, δ, α) if, for every subgraph G[S ] induced by S ⊂ V with |S | > η|V |, we have
(α − δ)
|S | 2
|S | < E (G[S ]) < (α + δ) . 2
Rödl [13] (see also [7]) answered affirmatively a question posed by Erdős (see, e.g., [5,11] and [1, p. 363]) by proving the following result. Theorem 1.1. For all integers k > 1 and every α , η ∈ (0, 1), there exist δ > 0 and n0 such that the following holds for all integer n > n0 .
E-mail addresses: [email protected] (Y. Kohayakawa), [email protected] (G.O. Mota), [email protected] (M. Schacht), [email protected] (A. Taraz). http://dx.doi.org/10.1016/j.ejc.2017.04.007 0195-6698/© 2017 Elsevier Ltd. All rights reserved.
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Every n-vertex graph G that satisfies Q(η, δ, α) contains all graphs with k vertices as induced subgraphs. The quantification in Theorem 1.1 is what makes it unexpected. Indeed, note that η is not required to be small; it is allowed to be any constant less than 1. We establish a variant of Theorem 1.1 in the context of sparse 3-uniform hypergraphs. Our main result (see Theorem 1.2) allows one to count the number of copies (not necessarily induced) of certain fixed 3-uniform hypergraphs H in spanning subgraphs G of sparse jumbled 3-uniform hypergraphs Γ . Jumbled graphs are well studied due to the fruitful work of Thomason [14,15] and others (see also [2–4,9]). For 3-uniform hypergraphs we shall consider the following notion here. Let Γ = (V , E ) be a 3-uniform hypergraph and let X ⊂
V 2
and Y ⊂ V be given. Denote by EΓ (X , Y ) the set of pairs
({x1 , x2 }, y) ∈ X × Y such that {x1 , x2 , y} ∈ E and set eΓ (X , Y ) = |EΓ (X , Y )|. We say that Γ is (p, β)-jumbled if, for all subsets X ⊂
V 2
and Y ⊂ V , we have
eΓ (X , Y ) − p|X | |Y | 6 β |X | |Y |. A hypergraph H is called linear if every pair of edges shares at most one vertex. A hyperedge e of a linear 3-uniform hypergraph H is a connector if there exist a vertex v ̸∈ e and hyperedges e1 , e2 , and e3 each containing v and each intersecting e in exactly one vertex. Note that in the context of graphs the analogous definition would result in an edge e that intersects two edges e1 and e2 that a share vertex v outside e; in other words, the edges e, e1 , and e2 form a graph triangle. Our result can be viewed as a counting lemma for fixed linear, connector-free 3-uniform hypergraphs H contained in certain spanning 3-uniform subhypergraphs Gn of (p, o(p2 n3/2 ))-jumbled hypergraphs Γ for sufficiently large p and n. In comparison to Theorem 1.1 one may think of Γ in our result as replacing the complete graph Kn , which contains the spanning graph G satisfying a ‘‘certain’’ condition Q(η, δ, α) on the distribution of its edge set. Below we define the variant of Q(η, δ, α) for 3-uniform hypergraphs that is imposed on Gn . We say that a 3-uniform hypergraph G = (V , E ) satisfies property Q′ (η, δ, q) if, for all X ⊂
and Y ⊂ V with |X | > η
|V | 2
V 2
and |Y | > η|V |, we have
(1 − δ)q|X | |Y | 6 EG (X , Y ) 6 (1 + δ)q|X | |Y |. Considering the cardinality of EG (X , Y ) for certain X ⊂
(1)
V 2
and Y ⊂ V to obtain information on
the subhypergraphs of G has recently been shown to be fruitful (see [12]). Since we consider sparse 3-uniform hypergraphs here, we also require a boundedness condition. Given a pair {v1 , v2 } ∈
V 2
,
we define NG ({v1 , v2 }) = v3 ∈ V : {v1 , v2 , v3 } ∈ E
and we say that G = (V , E ) satisfies BDD(k, C , q) if, for all 1 6 r 6 k and for all distinct 2-element
sets S1 , . . . , Sr ∈
V 2
, we have
|NG (S1 ) ∩ · · · ∩ NG (Sr )| 6 Cnqr . Finally, we can state the variant of Theorem 1.1 for 3-uniform hypergraphs considered here. In what follows, given hypergraphs H and G, we let Emb(H , G) be the set of embeddings of H into G, i.e., the set of edge preserving injective maps ϕ : V (H ) → V (G). Theorem 1.2. For all constants ε , α , η ∈ (0, 1) and C > 1, and every integer k > 5, there exist δ , γ > 0 such that if p = p(n) ≫ n−2/(k−1) and p = p(n) = o(1) and n is sufficiently large, then the following holds for every α p 6 q 6 p and every β 6 γ p2 n3/2 . Suppose that
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(i) Γ = (V , EΓ ) is an n-vertex (p, β)-jumbled 3-uniform hypergraph and n (ii) G = (V , EG ) is a spanning subhypergraph of Γ with |EG | = q 3 and G satisfies Q′ (η, δ, q) and BDD(k, C , q). Then for every linear 3-uniform connector-free hypergraph H on k vertices we have
|Emb(H , G)| − nk q|E (H )| < εnk q|E (H )| . We close this introduction with a few remarks concerning Theorem 1.2. Note that if p ≫ n−1/4 , then the random 3-uniform hypergraph, where each possible edge exists with probability p independently of all other edges, is (p, γ p2 n3/2 )-jumbled with high probability for all γ > 0. Therefore, our result applies to appropriate spanning subhypergraphs of such random 3-uniform hypergraphs. The hypergraph H in Theorem 1.2 is required to be linear and connector-free. It is possible that both conditions could be further relaxed. In fact, in the context of dense hypergraphs the assumption imposed on the edge distribution of G from (1) yields the right number of embeddings for a slightly larger class (see, e.g., [10,16] for details). Also it might be possible that the condition that H should be connector-free is not necessary. We inherit this condition from a counting lemma proved in [6] (see Lemma 3.1) and this is the only reason why we impose it here. Most of the definitions generalise naturally to r-uniform hypergraphs for r larger than 3, and it would be interesting to extend Theorem 1.2 to r-uniform hypergraphs with r > 3. Unfortunately such a generalisation presents new difficulties and we leave this open here. Organisation. We give an overview of the proof of Theorem 1.2 in Section 2. The proof will be based on some intermediate lemmas, which we establish in Section 3. In Section 4 we present the proof of Theorem 1.2, which, after all the work in Section 3, will be quite short. 2. Overview of the proof The first step in the proof of Theorem 1.2 is to show that, under the conditions of the theorem, G satisfies a strong property involving degrees and co-degrees (see Lemmas 2.4, 3.3 and 3.4). After that we apply a counting result (Lemma 3.1). Before we discuss the scheme of the proof, let us define some hypergraph properties, called Discrepancy, Pair, and Tuple. Definition 2.1 (DISC—Discrepancy Property). Let G = (V , E ) be a 3-uniform hypergraph and let X , Y ⊂ V be given. We say that the pair (X , Y ) satisfies DISC(q, p, ε) in (G or (X , Y )G satisfies DISC(q, p, ε)) if for all X ′ ⊂
X 2
and Y ′ ⊂ Y we have
eG (X ′ , Y ′ ) − q|X ′ | |Y ′ | 6 εp |X | |Y |. 2
Furthermore, if (V , V ) satisfies DISC(q, p, ε) in G, then we say that the hypergraph G satisfies DISC(q, p, ε). For a 3-uniform hypergraph G = (V , E ), a set of vertices Y ⊂ V , and pairs S1 , S2 ∈
V 2
we denote
NG (S1 ) ∩ Y by NG (S1 ; Y ) and NG (S1 ) ∩ NG (S2 ) ∩ Y by NG (S1 , S2 ; Y ). Definition 2.2 (PAIR—Pair Property). Let G = (V , E ) be a 3-uniform hypergraph and let X , Y ⊂ V be given. We say that the pair (X , Y ) satisfies PAIR(q, p, δ) in (G or (X , Y )G satisfies PAIR(q, p, δ)) if the following conditions hold:
|NG (S1 ; Y )| − q|Y | 6 δ p |X | |Y |, X
2
S1 ∈ 2
X
S1 ∈ 2
2 |NG (S1 , S2 ; Y )| − q2 |Y | 6 δ p2 |X | |Y |. X
S2 ∈ 2
2
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Furthermore, if (V , V ) satisfies PAIR(q, p, δ) in G, then we say that the hypergraph G satisfies PAIR(q, p, δ). Definition 2.3 (TUPLE—Tuple Property). We define TUPLE(δ, q) as the family of n-vertex 3-uniform hypergraphs G = (V , E ) such that the following two conditions hold: (i) |NG (S1 )| − nq < δ nq for all but at most δ
n 2
sets S1 ∈
V 2
;
n V (ii) |NG (S1 ) ∩ NG (S2 )| − nq2 < δ nq2 for all but at most δ ( 22 ) pairs {S1 , S2 } of distinct sets in 2 . The next result allows us to obtain property TUPLE from PAIR. Since its proof is simple we will omit it. Lemma 2.4. For all 0 < α 6 1 and 0 < δ < 1 there exists δ ′ > 0 such that if a 3-uniform hypergraph G satisfies PAIR(q, p, δ ′ ) for α p 6 q 6 p, then G satisfies TUPLE(δ, q). In what follows we explain the organisation of the proof. Consider the setup of Theorem 1.2. In order to obtain the conclusion of the theorem, we will use a counting result (Lemma 3.1), which requires that G satisfies properties BDD and TUPLE for the appropriate parameters. Since G satisfies BDD by hypothesis, it suffices to prove that G satisfies TUPLE. Using Lemma 3.3 it is possible to obtain DISC from property Q′ . Then, using that G satisfies DISC one can show that G satisfies PAIR using Lemma 3.4, which implies TUPLE by Lemma 2.4. The quantification used in these implications is carefully analysed in Section 4. 3. Main lemmas We start by stating the counting lemma needed in the proof of Theorem 1.2. In order to apply it to a 3-uniform n-vertex hypergraph G, we shall prove that G satisfies TUPLE(δ, q) for a sufficiently small δ and sufficiently large 0 < q = q(n) 6 1. Since Lemma 2.4 allows us to obtain TUPLE from PAIR, we need to proof that G satisfies PAIR(q, p, δ ′ ) for a sufficiently small δ ′ and appropriate functions p and q. This is done using Lemmas 3.3 and 3.4, which are proved, respectively, in the Sections 3.1 and 3.2. Given a 3-uniform hypergraph H, we define parameters dH = max{δ(J ) : J ⊂ H } and DH = min{3dH , ∆(H )}, where δ(J ) and ∆(J ) stand, respectively, for the minimum and maximum degree of a vertex in V (J ). The following result, proved in [6], is our counting lemma. Lemma 3.1. Let k > 5 be an integer and let ε > 0, C > 1 and an integer d > 2 be fixed. Let H be a linear 3-uniform connector-free hypergraph on k vertices such that DH 6 d. Then, there exists δ > 0 for which the following holds for any q = q(n) with q ≫ n−1/d and q = o(1 )n and for sufficiently large n. If G is an n-vertex 3-uniform hypergraph with |E (G)| = q 3 hyperedges and G satisfies BDD(DH , C , q) and TUPLE(δ, q), then
|Emb(H , G)| − nk q|E (H )| < εnk q|E (H )| . Recall that p ≫ n−2/(k−1) is one of the conditions in Theorem 1.2. Under this condition on p, since DH 6 (k − 1)/2 for any linear 3-uniform hypergraph H on k vertices, we can apply Lemma 3.1 with any q = q(n) such that q > α p for some 0 6 α 6 1. 3.1. Q′ implies DISC Given a 3-uniform hypergraph G = (V , E ) and subsets A ⊂
V 2
and non-empty B ⊂ V , the
q-density between A and B is defined as dq (A, B) =
|EG (A, B)| . q|A| |B|
Before we state the main result of this subsection, Lemma 3.3, we shall prove the following result, which is inspired by a result in [13] for graphs.
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Lemma 3.2. For all 0 < η < 1 and 0 < ε ∗ < (1 − η)/3, there exists δ > 0 such that, if G = (V , E ) is ′ an n-vertex 3-uniform hypergraph that satisfies Q (η, δ, q), then the following holds. For every C ⊂
V 2
and D ⊂ V such that |C | is a multiple of ⌈ε ∗
n ⌉ and |D| is a multiple of ⌈ε ∗ n⌉, 2
we have 1 − ε ∗ < dq (C , D) < 1 + ε ∗ . Proof. Fix η > 0 and 0 < ε ∗ < (1 − η)/3. Let δ = ε ∗ 3 /24 and put t = ⌈1/ε ∗ ⌉. Suppose G = (V , E ) is an n-vertex 3-uniform hypergraph that satisfies Q′ (η, δ, q). Now, fix C ⊂
V 2
and D ⊂ V such
⌉ and |D| = k2 ⌈ε n⌉ for some positive integers k1 and k2 . Let C1 , . . . , Ck1 and n ⌉ and |D1 | = 2 V ∗ · · · = |Dk2 | = ⌈ε n⌉. Now we partition the sets 2 \ C and V \ D, respectively, in sets Ck1 +1 , . . . , Ct n and Dk2 +1 , . . . , Dt such that |Ck1 +1 | = · · · = |Ct −1 | = ⌈ε ∗ 2 ⌉ and |Dk2 +1 | = · · · = |Dt −1 | = ⌈ε ∗ n⌉. n Note that |Ct | 6 ε ∗ 2 and |Dt | 6 ε ∗ n. We divide the rest of the into two parts. First, we prove that for any triple i, j, j′ ∈ [t − proof 1], |e(Ci , Dj ) − e(Ci , Dj′ )| 6 6δ 2n nq, and for any triple i, i′ , j ∈ [t − 1], |e(Ci , Dj ) − e(Ci′ , Dj )| 6 6δ 2n nq. To finish the proof we put these estimates together to show that 1 − ε ∗ < dq (C , D) < 1 + ε ∗ . ∗ Put X = C2 ∪ · · · ∪ Ct and Y = D3 ∪ · · · ∪ Dt . Since ε < (1 − η)/3, we have |X | = (t − 2)⌈ε ∗ 2n ⌉ + |Ct | > (t − 2)ε∗ 2n > η 2n and |Y | = (t − 3)⌈ε ∗ n⌉ + |Dt | > (t − 3)ε∗ n > ηn. Therefore, using Q′ (η, δ, q) and |D1 | = |D2 |, the following two inequalities hold. e(X , D1 ∪ Y ) − e(X , D2 ∪ Y ) 6 2δ|X |(|D1 | + |Y |)q, (2) e(C1 ∪ X , Y ) e(C1 ∪ X , Dj ∪ Y ) (3) (|C | + |X |)|Y |q − (|C | + |X |)(|D | + |Y |)q 6 2δ, for j ∈ {1, 2}. 1 1 j that |C | = k1 ⌈ε
∗ n
∗
2
D1 , . . . , Dk2 be, respectively, partitions of C and D such that |C1 | = · · · = |Ck1 | = ⌈ε ∗
Now we define the following for j ∈ {1, 2} p1j =
e(C1 ∪ X , Y ) + e(X , Dj ) e(C1 ∪ X , Y ) − . (|C1 | + |X |)|Y |q (|C1 | + |X |)(|Dj | + |Y |)q
By (3), the following holds for j ∈ {1, 2} e(C1 , Dj ) 6 p1j + 2δ. (4) (|C1 | + |X |)(|Dj | + |Y |)q Note that e(X , D1 )− e(X , D2 ) = e(X , D1 ∪ Y )− e(X , D2 ∪ Y ). Thus, using (2), we obtain the following p1j − 2δ 6
inequality.
|p11 − p12 | =
e(X , D1 ) − e(X , D2 )
|X | 6 2δ < 2δ. (|C1 | + |X |)(|D1 | + |Y |)q |C1 | + |X |
(5)
Putting (4) and (5) together, we obtain the following inequality.
|e(C1 , D1 ) − e(C1 , D2 )| < 6δ(|C1 | + |X |)(|D1 | + |Y |)q < 6δ
n 2
nq.
Applying the same strategy one can prove that, for any triple i, j, j′ ∈ [t − 1],
|e(Ci , Dj ) − e(Ci , Dj′ )| < 6δ
n 2
nq.
(6)
Analogously, we obtain the following equation for any triple i, i′ , j ∈ [t − 1].
|e(Ci , Dj ) − e(Ci′ , Dj )| < 6δ
n 2
nq.
(7)
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By (6) and (7), we have |e(Ci , Dj ) − e(Ci′ , Dj′ )| < 12δ
|dq (Ci , Dj ) − dq (Ci′ , Dj′ )| <
12δ
n 2
nq
|Ci | |Dj |q
<
n
12δ
(ε∗ )2
2
=
)
–
nq for any i, i′ , j, j′ ∈ [t − 1]. Therefore,
ε∗
(8)
2
holds for any i, i′ , j, j′ ∈ [t − 1]. Put WC = C1 ∪· · ·∪ Ct −1 and WD = D1 ∪· · ·∪ Dt −1 . Since |WC | > η and |WD | > ηn, we know, by property Q′ (η, δ, q), that 1 − δ < dq (WC , WD ) < 1 + δ.
n 2
(9)
Suppose for a contradiction that there exist indexes i0 , j0 ∈ [t − 1] such that either dq (Ci0 , Dj0 ) > 1 + ε ∗ or dq (Ci0 , Dj0 ) < 1 − ε ∗ . Then, by (8), either for all i, j ∈ [t − 1] we have dq (Ci , Dj ) > 1 + ε ∗ /2 or for all i, j ∈ [t − 1] we have dq (Ci , Dj ) < 1 − ε ∗ /2. But note that
dq (WC , WD ) =
i,j∈[t −1]
dq (Ci , Dj )|Ci | |Dj |q
.
|WC | |WD |q
Then, either
(t − 1)2 (1 − ε ∗ /2) ε ∗ 2n ⌈ε ∗ n⌉q = (1 − ε ∗ /2) < 1 − δ, dq (WC , WD ) < |WC | |WD |q or
(t − 1)2 (1 + ε ∗ /2) ε ∗ 2n ⌈ε ∗ n⌉q dq (WC , WD ) > = (1 + ε ∗ /2) > 1 + δ, |WC | |WD |q a contradiction with (9). Therefore, for all i, j ∈ [t − 1], 1 − ε ∗ < dq (Ci , Dj ) < 1 + ε ∗ .
(10)
It remains to estimate the densities dq (Ck1 , Dj ) and dq (Ci , Dk2 ) with k1 = t and k2 = t for all 1 6 i 6 k1 n n and 1 6 j 6 k2 . Note that k1 = t and k2 = t imply, respectively, ⌈ε ∗ 2 ⌉ = ε ∗ 2 and ⌈ε ∗ n⌉ = ε ∗ n, but in these cases one can prove in the same way we proved (10). Therefore, putting all these estimates together, we obtain 1 − ε ∗ < dq (C , D) < 1 + ε ∗ . The next lemma shows how one can obtain discrepancy properties from Q′ in spanning subhypergraphs of sufficiently jumbled 3-uniform hypergraphs. Lemma 3.3. For all 0 < ε ′ , η, σ < 1 there exists δ > 0 such that for every α > 0 there exists γ > 0 such that the following holds. Let Γ = (V , EΓ ) be an n-vertex (p, β)-jumbled 3-uniform hypergraph for 0 < p = p(n) 6 1 such that α p 6 q 6 p and β 6 γ pn3/2 . Let G = (V , EG ) be a spanning subhypergraph of Γ . If G satisfies Q′ (η, δ, q), then every pair (X , Y )G with X , Y ⊂ V such that |X |, |Y | > σ n satisfies DISC(q, p, ε ′ ). Proof. Fix ε ′ , η, σ > 0 and let ε ∗ = min ε ′ σ 2 /24, (1 − η)/4 . Let δ ′ be the constant given by Lemma 3.2 applied with η and ε ∗ . Put δ = min{δ ′ , ε ′ }, α > 0 and γ = σ 3/2 αε ′ /2. Suppose that α p 6 q 6 p and β 6 pγ n3/2 . Let Γ = (V , EΓ ) be an n-vertex (p, β)-jumbled 3-uniform hypergraph and let G = (V , EG ) be a spanning subhypergraph of Γ such that G satisfies Q′ (η, δ, q). Let (X , Y )G be a pair with X , Y ⊂ V such that |X |,|Y | > σ n. We want to prove that
2
(X , Y )G satisfies DISC(q, p, ε′ ). For this, fix arbitrary subsets X ′ ⊂ X2 and Y ′ ⊂ Y . We will prove that eG (X ′ , Y ′ ) − q|X ′ | |Y ′ | 6 ε ′ p |X | |Y |. 2 |X | Upper bound. First, consider the case where |X ′ | 6 ε ′ 2 or |Y ′ | 6 ε ′ |Y |. Note that, from the choice of γ and β , since |X |, |Y | > σ n, we have |X | |Y |. (11) β |X ′ | |Y ′ | 6 αε ′ p 2
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Therefore, eG (X ′ , Y ′ ) 6 p|X ′ | |Y ′ | + β
|X ′ | |Y ′ |
6 q|X ′ | |Y ′ | + (1 − α)p|X ′ | |Y ′ | + β
|X ′ | |Y ′ | |X | 6 q|X ′ | |Y ′ | + (1 − α)pε ′ |Y | + β |X ′ | |Y ′ | 2 | X | 6 q|X ′ | |Y ′ | + ε ′ p |Y |,
(12)
2
where the first inequality follows from the jumbledness of Γ and the fact that G is a subhypergraph of
Γ , the second one follows from the value of q, the third one follows from the fact that |X ′ | 6 ε ′ |X2 | |X | or |Y ′ | 6 ε ′ |Y |, and the last one is a consequence of (11). Thus, we may assume |X ′ | > ε ′ 2 and
|Y ′ | > ε′ |Y |. We consider four the size of |X ′ | and |Y ′ |. cases, ′depending on n n ′ ∗ ∗ Case 1: (|X | > (1 − ε ) 2 and |Y | > (1 − ε )n). By the choice of ε ∗ , we have |X ′ | > η 2 and ′ ′ |Y | > ηn. By Q (η, δ, q) we conclude that |X | ′ ′ ′ ′ ′ ′ ′ eG (X , Y ) 6 (1 + δ)q|X | |Y | 6 q|X | |Y | + ε p |Y |. 2
Case 2: (|X ′ | < (1 − ε ∗ )
n 2
and |Y ′ | < (1 − ε ∗ )n). Note that, since |X ′ | < (1 − ε ∗ )
|Y | < (1 − ε )n, there exist subsets X ⊂ ′
∗
V 2
∗
∗
∗
′
′′
∗
n 2
and
′
and Y ⊂ V such that X = X ∪ X and Y = Y ∪ Y ′′ ,
n n with X ∩ X = ∅ and Y ∩ Y = ∅, where |X | 6 ε ∗ 2 and |X ∗ | is multiple of ⌈ε ∗ 2 ⌉, and |Y ′′ | 6 ε ∗ n ∗ ∗ and |Y | is a multiple of ⌈ε n⌉. Then, we can use Lemma 3.2 to obtain the following inequality.
′
′′
′
′′
′′
eG (X ′ , Y ′ ) 6 eG (X ∗ , Y ∗ ) 6 (1 + ε ∗ )|X ∗ | |Y ∗ |q 6 (1 + ε ∗ )q|X ′ | |Y ′ | + 2q |X ′ | |Y ′′ | + |X ′′ | |Y ′ | + |X ′′ | |Y ′′ | .
Since ε ∗ 6 ε ′2 σ 2 /16, we have |X ′′ | 6 ε ∗
n 2
6 (ε ′ /8)|X ′ | and |Y ′′ | 6 ε ∗ n 6 (ε ′ /8)|Y ′ |. Therefore,
eG (X ′ , Y ′ ) 6 (1 + ε ∗ )q|X ′ | |Y ′ | + 2q 3(ε ′ /8)|X ′ | |Y ′ |
6 q|X ′ | |Y ′ | +
ε′ 4
q|X ′ | |Y ′ | +
3ε ′ 4
q|X ′ | |Y ′ | 6 q|X ′ | |Y ′ | + ε ′ p
|X |
2
|Y |.
Case 3: (|X ′ | > (1 − ε ∗ ) 2 and |Y ′ | < (1 − ε ∗ )n). As noticed before, since |Y ′ | < (1 − ε ∗ )n, there exist subsets Y ∗ , Y ′′ ⊂ V such that Y ∗ = Y ′ ∪ Y ′′ with Y′ ∩ Y ′′ = ∅, where |Y ′′ | 6 ε ∗ n and |Y ∗ | is a
n
multiple of ⌈ε ∗ n⌉. Note that there exist subsets X˜ , X ′′ ⊂
V such that X ′ = X˜ ∪ X ′′ with X˜ ∩ X ′′ = ∅, 2 n where |X ′′ | 6 ε 2 and |X˜ | is a multiple of ⌈ε ∗ 2 ⌉. ′′ ′′ ′′ then we ‘‘complete’’ X ′′ with elements of If X is empty, then put W = ∅. If X is not empty, V ′′ ′′ ′′ ′′ ∗ n to obtain W such that X ⊂ W and |W | = ⌈ε 2 ⌉ (note that possibly W ′′ ∩ X˜ ̸= ∅). Thus, 2 |X˜ | + |W ′′ | 6 |X ′ | + ε ∗ 2n . By using Lemma 3.2, we have
∗ n
eG (X ′ , Y ′ ) 6 eG (W ′′ , Y ∗ ) + eG (X˜ , Y ∗ )
6 (1 + ε ∗ )q |Y ∗ | |W ′′ | + |Y ∗ | |X˜ |
= (1 + ε ∗ )q |Y ′ | |W ′′ | + |Y ′′ | |W ′′ | + |Y ′ | |X˜ | + |Y ′′ | |X˜ | n n 6 (1 + ε ∗ )q |Y ′ | |X ′ | + ε ∗ + |Y ′′ | |X ′ | + ε ∗ 2n 2 ∗ ′ ′ ∗ ∗ n ∗ ′ ′ 6 (1 + ε )q|X | |Y | + 2q ε |Y | + |X |ε n + ε ε∗ n . 2
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Since ε ∗ 6 ε ′2 σ 2 /16, we have ε eG (X ′ , Y ′ ) 6 q|X ′ | |Y ′ | +
∗ n
ε′ 4
6 q|X ′ | |Y ′ | + ε ′
Case 4: (|X ′ | < (1 − ε ∗ )
n
Lower bound. If |X ′ | 6 ε ′
2
2
|X |
2
X˜ ⊂
3ε ′ 8
|X ′ | |Y ′ |
|Y |p.
and |Y ′ | > (1 − ε ∗ )n). This case is analogous to Case 3.
|X |
2
or |Y ′ | 6 ε ′ |Y |, then there is nothing to prove, because ε ′
q|X | |Y |. Therefore, assume that |X | > ε ′
–
6 (ε ′ /8)|X ′ | and ε ∗ n 6 (ε ′ /8)|Y ′ |. Therefore,
q|X ′ | |Y ′ | + 2q
)
′
′
′
|X | 2
|X | 2
|Y |p >
and |Y | > ε |Y |. Clearly, there exist subsets ′
′
V 2
, Y˜ ⊂ V such that X ′ = X˜ ∪ X ′′ , Y ′ = Y˜ ∪ Y ′′ , with X˜ ∩ X ′′ = Y˜ ∩ Y ′′ = ∅, where |X ′′ | 6 ε ∗ 2n , |X˜ | is a multiple of ⌈ε ∗ 2n ⌉, |Y ′′ | 6 ε ∗ n and |Y˜ | is a multiple of ⌈ε ∗ n⌉. Since ε ∗ 6 ε ′2 σ 2 /8, we have n |X ′′ | 6 ε∗ 6 (ε ′ /4)|X ′ | 6 ε ′ /4(1 − ε ∗ ) |X ′ | 2
and |Y | 6 ε n 6 (ε ′ /4(1 − ε ∗ ))|Y ′ |. Then, by Lemma 3.2, since eG (X ′ , Y ′ ) > eG (X˜ , Y˜ ), we have ′′
∗
eG (X ′ , Y ′ ) > (1 − ε ∗ )|X˜ | |Y˜ |q
= (1 − ε ∗ )q |X ′ | |Y ′ | − |X ′ | |Y ′′ | − |X ′′ | |Y ′ | + |X ′′ | |Y ′′ | > (1 − ε ∗ )q|X ′ | |Y ′ | − (1 − ε ∗ )q |X ′ | |Y ′′ | + |X ′′ | |Y ′ | > q|X ′ | |Y ′ | − ε ∗ q|X ′ | |Y ′ | − (1 − ε ∗ )q (ε ′ /2(1 − ε ∗ ))|X ′ | |Y ′ | > q|X ′ | |Y ′ | −
ε′ 2
> q|X ′ | |Y ′ | − ε ′
q|X ′ | |Y ′ | −
|X | 2
ε′ 2
q|X ′ | |Y ′ |
|Y |p.
3.2. DISC implies PAIR The next lemma, which is a variation of Lemma 9 in [8], makes it possible to obtain PAIR from DISC in spanning subhypergraphs of sufficiently jumbled 3-uniform hypergraphs. Lemma 3.4. For all 0 < α 6 1 and δ ′ > 0 there exists ε ′ > 0 such that for all σ > 0 there exist γ > 0 such that the following holds for sufficiently large n. Suppose that
√
(i) Γ = (V , EΓ ) is an n-vertex 3-uniform (p, β)-jumbled hypergraph with p > 1/ n, (ii) G = (V , EG ) is a spanning subhypergraph of Γ , and (iii) X , Y ⊂ V with |X |, |Y | > σ n. Then, the following holds. If β 6 γ p2 n3/2 and (X , Y )G satisfies DISC(q, p, ε ′ ) for some q with α p 6 q 6 p, then (X , Y )G satisfies PAIR(q, p, δ ′ ). We need the following results in order to prove Lemma 3.4. First, consider the following fact, which is similar to [8, Fact 13]. Fact 3.5. Let Γ be a 3-uniform (p, β)-jumbled hypergraph. Let U ⊂
V 2
and W ⊂ V and ξ > 0. If we
have |NΓ ({x, y}, W )| > (1 + ξ )p|W | for every {x, y} ∈ U or we have |NΓ ({x, y}, W )| 6 (1 − ξ )p|W |
Y. Kohayakawa et al. / European Journal of Combinatorics (
)
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for every {x, y} ∈ U, then
|U | |W | 6
β2 . ξ 2 p2
Proof. Let Γ , U, W and ξ be as in the statement and suppose that for every {x, y} ∈ U we have NΓ ({x, y}, W ) > (1 + ξ )p|W |. Suppose for a contradiction that |U | |W | > β2 22 . Then, eΓ (U , W ) > ξ p √ |U |(1 + ξ )p|W | > p|U | |W | + β |U | |W |, a contradiction to the jumbledness of Γ . The case where |NΓ ({x, y}, W )| 6 (1 − ξ )p|W | for every {x, y} ∈ U is analogous. Our next result, Lemma 3.8, is very similar to [8, Lemma 21], but in Lemma 3.8 we consider bipartite
, V ; EΓ instead of Γ = (U , V ; EΓ ) in [8], and we consider subsets X1 , X2 of V2 n with |X1 |, |X2 | > η 2 instead of subsets X1 , X2 of V with |X1 |, |X2 | > ηn. Due to this fact, the value of β in Lemma 3.8 is γ p2 n3/2 , while in [8, Lemma 21] we have β = γ pn. The proof of Lemma 3.8 is
graphs Γ =
V 2
identical to the proof of [8, Lemma 21] and we omit it here. Let Γ = (V , EΓ ) be a graph and let X , Y ⊂ V . As usual, we denote by eΓ (X , Y ) the number of edges of Γ with one end-vertex in X and one end-vertex in Y , where edges contained in X ∩ Y are counted twice. We need to define jumbledness and discrepancy for graphs. Definition 3.6 (Jumbledness for Graphs). We say that Γ = √(V , EΓ ) is a (p, β)-jumbled graph if, for all subsets X , Y ⊂ V , we have eΓ (X , Y ) − p|X | |Y | 6 β |X | |Y |. Furthermore, a bipartite graph
ΓB = (U , V ; E ) is called (p, β)-jumbled if, for all X ⊂ U and Y ⊂ V , we have eΓ (X , Y ) − p|X | |Y | 6 √ β |X | |Y |.
Definition 3.7 (Discrepancy for Graphs). Let G = (V , E ) be a graph and let X , Y ⊂ V be disjoint. We say that (X , Y ) satisfies DISC(q, p, ε) in G (or (X , Y )G satisfies DISC(q, p, ε)) if for all X ′ ⊂ X and Y ′ ⊂ Y we have
eG (X ′ , Y ′ ) − q|X ′ | |Y ′ | 6 ε p|X | |Y |. Lemma 3.8. For all positive real ϱ0 and ν , there exists a positive real µ such that, for all σ ′ > 0, there exist γ > 0 and n0 > 0 such that for all n > n0 , the following holds. Suppose
√ , V ; EΓ is a bipartite (p, β)-jumbled graph with |V | > n, p > 1/ n and β 6 γ p2 n3/2 , n V (ii) X1 , X2 ⊂ 2 and Y ⊂ V with |X1 |, |X2 | > σ ′ 2 , |Y | > σ ′ n, (iii) B = (X1 , X2 ; EB ) is an arbitrary bipartite graph. (i) Γ =
V 2
Then, if (X1 , X2 )B satisfies DISC(ϱ, 1, µ) for some ϱ with ϱ0 6 ϱ 6 1, then for all but at most ν|Y | vertices y ∈ Y , the pair NΓ (y, X1 ), NΓ (y, X2 ) B satisfies DISC(ϱ, 1, ν). We need two more facts before proving of Lemma 3.4. Fact 3.9 ([8, Fact 22]). Suppose ϱ0 > 0, µ > 0 and B = (X , EB ) is a graph with |EB | > ϱ0 there exist disjoint subsets X1 , X2 ⊂ X such that
|X | 2
. Then
(i) (X1 , X2 )B satisfies DISC(ϱ, 1, µ) for some ϱ > ϱ0 ,
(ii) |X1 |, |X2 | > ζ |X | for ζ = ϱ0 100/µ /4. 2
Fact 3.10. Let Γ = (V , E ) be a 3-uniform hypergraph and let Γ ′ = where E ′ = {{v1 , v2 }, v} : {v1 , v2 } ∈
and only if Γ ′ is (p, β)-jumbled.
V 2
V 2
, V ; E ′ be a bipartite graph,
, v ∈ V and {v1 , v2 , v} ∈ E . Then, Γ is (p, β)-jumbled if
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)
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We have stated all the tools needed in the proof of Lemma 3.4. This proof is very similar to the proof of [8, Lemma 9]. Proof of Lemma 3.4. Let 0 < α 6 1 and 0 < δ ′ < 1 be given. Put ξ = δ ′ /6, ϱ0 = δ ′ /50 and ν = α 2 ξ ϱ0 /64. Let µ be obtained by an application of Lemma 3.8 with parameters ϱ0 and ν . Without 100/µ2 loss of generality, assume µ < ξ ϱ0 /4. Let ζ = ϱ0 /4 be given and put ε ′ = min αδ ′2 /36, (α 3 ξ ϱ0 ζ /64)2 . Now fix σ > 0 and let σ ′ = ζ σ 2 /2. Following the quantification of Lemma 3.8 applied with parameter σ ′ we obtain γ ′ and n0 . Then, put γ = min γ ′ , σ 3 δ ′ /12, (α/2) ξ ϱ0 σ σ ′ /24 . √ Finally, consider n sufficiently large and suppose p > 1/ n. 2 3/2 Fix β 6 γ p n and consider a 3-uniform (p, β)-jumbled hypergraph Γ = (V , EΓ ) such that |V | = n and let G = (V , EG ) be a spanning subhypergraph of Γ . Let X , Y be subsets of V such that ′ |X |, |Y| > σ n. Suppose that (X , Y )G satisfies DISC(q, p, ε ) for some q with α p 6 q 6 p, i.e., for all X X ′ ⊂ 2 and Y ′ ⊂ Y the following holds.
eG (X ′ , Y ′ ) − q|X ′ | |Y ′ | 6 ε ′ p |X | |Y |.
(13)
2
We want to prove that the following inequalities hold:
|NG (S1 ; Y )| − q|Y | 6 δ ′ p |X | |Y |,
(14)
2
X
S1 ∈ 2
X
S1 ∈ 2
2 |NG (S1 , S2 ; Y )| − q2 |Y | 6 δ ′ p2 |X | |Y |.
(15)
2
X
S2 ∈ 2
We start by verifying (14). For at most δ ′
|X |
2
/6 pairs S ∈
X 2
, we have
|NG (S , Y )| − q|Y | > (δ ′ /3)q|Y |.
with at least δ ′
/12 elements such that, for all {x, x } ∈ BX , either |NG ({x, x }, Y )| > (1 + δ /3)q|Y | or for all of them we have |NG ({x, x′ }, Y )| < (1 − δ ′ /3)q|Y |. In either case, we would have ′2 ′2 eG (BX , Y ) − q|BX | |Y | > δ q |X | |Y | > δ α p |X | |Y | > ε ′ p |X | |Y |, Indeed, otherwise there would be a set BX ⊂ ′
′
X 2
|X | 2
′
36
36
2
2
2
where the last inequality follows from the choice of ε . But this contradicts (13) when we put X ′ = BX and Y ′ = Y . X Let U be the set of pairs S ∈ 2 such that |NΓ (S , Y )| > 2p|Y |. By Fact 3.5 applied with U and Y ′
with ξ = 1, we know that there exist at most β 2 /p2 |Y | elements S ∈ U such that |NΓ (S , Y )| > 2p|Y |. Therefore,
′ ′ |NG (S , Y )| − q|Y | 6 |X | δ q|Y | + δ |X | 2p|Y | + (β 2 /p2 |Y |)|Y | 2
X
3
6
2
S∈ 2
6p
|X | 2
|Y |
2δ ′ 3
+ (β/p)2 6 δ ′ p
|X |
2
where the last inequality follows from the facts that β 6 γ p2 n3/2 and γ 6 that (14) holds.
|Y |, σ 3 δ ′ /12. We just proved
Y. Kohayakawa et al. / European Journal of Combinatorics (
)
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Suppose for a contradiction that (15) does not hold. Then,
2 |NG (S1 , S2 ; Y )| − q2 |Y | > δ ′ p2 |X | |Y |.
X
S1 ∈ 2
(16)
2
X
S2 ∈ 2
Define the following sets of ‘‘bad’’ pairs.
B1 = (S1 , S2 ) ∈ B2 = (S1 , S2 ) ∈
X
2 X
2
X
×
2
×
X
2
: |NΓ (S1 , Y )| > 2p|Y | , \ B1 : |NΓ (S1 , S2 , Y )| > 4p2 |Y | .
Since Γ is (p, β)-jumbled, it follows that
|B1 | 6
β2 2 p |Y |
|X | 2
6
γ 2 n3 p2 |Y |
|X |
6
2
γ 2 n2 p 2 σ
|X |
6
2
δ′ 3
p2
|X |
2
2
where the first inequality follows from Fact 3.5 applied to the sets
U =
X
S1 ∈
2
: |NΓ (S1 , Y )| > 2p|Y |
and Y with ξ = 1. The second inequality follows from thechoice of β , the third one follows from |Y | > σ n, and the last one holds because |X | > σ n and γ 6 σ 3 δ ′ /12. We want to bound |B2 | from above. By definition, if a pair of vertices belongs to B2 , then it X does not belong to B1 . Then, consider a pair of vertices S1 ∈ 2 such that |NΓ (S1 , Y )| 6 2p|Y |. ′ Consider a set Y ⊂ Y of size exactly 2p|Y | that contains NΓ (S1 , Y ). Applying Fact 3.5 to the sets
: |NΓ (S2 , Y ′ )| > 2p|Y ′ | and Y ′ with ξ = 1, we conclude that there are at most β 2 /(p2 |Y ′ |) X pairs S2 ∈ 2 such that |NΓ (S1 , S2 , Y )| > 4p2 |Y |. Therefore,
S2 ∈
X 2
2 γ 2 n2 p δ ′ |X | 6 p . |B2 | 6 2 2σ 6 2 2 X X The summation below is over the pairs (S1 , S2 ) ∈ 2 × 2 \ (B1 ∪ B2 ). By (16) and the upper
|X |
β2 6 p2 2p|Y |
|X |
bounds on B1 and B2 we conclude that
2 |NG (S1 , S2 ; Y )| − q2 |Y | > δ ′ p2 |X | |Y | − |B1 | |Y | − |B2 |2p|Y | 2
> δ ′ p2
=
δ′ 3
2
p
|X |
2
2
|X | 2
|Y | −
2δ ′ 3
p2
|X |
2
2
|Y |
2 |Y |.
(17)
The contribution of the pairs (S1 , S2 ) ̸∈ (B1 ∪ B2 ) with |NG (S1 , S2 ; Y )| − q2 |Y | 6 δ ′ q2 |Y |/6 to the sum in (17) is at most
δ′ 6
p2
|X | 2
2 |Y |.
Note that, by the definition of B2 , for all (S1 , S2 ) ̸∈ (B1 ∪ B2 ), the following holds.
|NG (S1 , S2 ; Y )| − q2 |Y | 6 max q2 |Y |, (4p2 − q2 )|Y | 6 4p2 |Y |.
(18)
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Hence, by (17) and (18), there exist at least δ ′
|X |
2
2
)
/24 pairs (S1 , S2 ) ∈
–
X 2
×
X 2
\ (B1 ∪ B2 ) such
that ′ |NG (S1 , S2 ; Y )| − q2 |Y | > δ q2 |Y | = ξ q2 |Y |.
(19)
6
Now let us define two auxiliary graphs B+ and B− with vertex-set
E (B+ ) =
E (B− ) =
X 2
{S1 , S2 } ∈
:
2
X 2
{S1 , S2 } ∈
Since there are at least δ ′
|X |
2
2
:
2
|X |
2
max{e(B+ ), e(B− )} >
4
X 2
and edge-sets as follows.
(1 + ξ )q2 |Y | < |NG (S1 , S2 ; Y )| 6 4p2 |Y | |NG (S1 , S2 ; Y )| < (1 − ξ )q2 |Y | .
/24 pairs (S1 , S2 ) ∈
2
δ ′ /24
−
|X |
2
X 2
×
X 2
> ϱ0
|X | 2
2
such that (19) holds, we have
,
where in the first inequality the term ‘‘4’’ in the denominator comes from the fact that now we are counting unordered pairs and the edges belong either to E (B+ ) or E (B− ). Furthermore, we discount the pairs {S1 , S1 }. Suppose without lost of generality that e(B+ ) > ϱ0
|X |
2
. Then, Fact 3.9 implies that there exist
2
subsets X1 , X2 ⊂ 2 with |X1 |, |X2 | > ζ 2 such that (X1 , X2 )B+ satisfies DISC(ϱ, 1, µ) for some ϱ > ϱ0 . Recall that Γ = (V , EΓ) isa 3-uniform (p, β)-jumbled hypergraph with n vertices. By Fact 3.10, X
the bipartite graph Γ ′ = ( EΓ ′
V 2
|X |
, V ; EΓ ′ ), where
= {v1 , v2 }, v : {v1 , v2 } ∈ V2 , v ∈ V and {v1 , v2 , v} ∈ EΓ
is a (p, β)-jumbled graph. Note that X1 , X2 ⊂
X 2
⊂
V 2
with |X1 |, |X2 | > ζ
|X |
2
> ζ
σn 2
> (ζ σ 2 /2) 2 > σ ′ 2 . Therefore, the hypotheses of Lemma 3.8 are satisfied. By Lemma 3.8 we conclude that for all but at most ν|Y | vertices y ∈ Y , the pair (NΓ ′ (y, X1 ), NΓ ′ (y, X2 ))B+ satisfies DISC(ϱ, 1, ν), which implies the following statement for all but at most ν|Y | vertices y ∈ Y .
n
n
(NΓ (y, X1 ), NΓ (y, X2 ))B+ satisfies DISC(ϱ, 1, ν).
(20)
Now let us estimate the number of triplets (S1 , S2 , y) in X1 × X2 × Y such that {S1 , S2 } is an edge of B+ and the pairs S1 and S2 belong to the neighbourhood of y in G. Formally, we define such triplets as follows.
T = {(S1 , S2 , y) ∈ X1 × X2 × Y : S1 ∈ NG (y, X1 ), S2 ∈ NG (y, X2 ), {S1 , S2 } ∈ E (B+ )}. By the definition of B+ we have
|T | > (1 + ξ )q2 |Y |eB+ (X1 , X2 ) > (1 + ξ )q2 |Y |(ϱ − µ)|X1 | |X2 | ξ > 1+ ϱq2 |X1 | |X2 | |Y |, 2
(21)
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)
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where in the second inequality we used the fact that (X1 , X2 )B+ ∈ DISC(ϱ, 1, µ) and the last one follows from the choice of µ. + NG (y, X1 ), NG (y, X2 ) . Now we give an upper bound on |T | contradicting (21). We have |T | = y∈Y eB Put Y ′ = y ∈ Y : dΓ (y, Xi ) 6 2p|Xi | for both i = 1, 2 .
By (20), for all but at most ν|Y | vertices y ∈ Y ′ we have eB+ (NG (y, X1 ), NG (y, X2 )) 6 ϱNG (y, X1 ) |NG (y, X2 )| + ν|NΓ (y, X1 )| |NΓ (y, X2 )|
6 ϱ dG (y, X1 )dG (y, X2 ) + 4ν p2 |X1 | |X2 |.
The last inequality follows from the fact that y ∈ Y ′ . Now we willbound the terms related to vertices in Y \ Y ′ . By Fact 3.5, we have |Y \ Y ′ | 6 β 2 1/p2 |X1 | + 1/p2 |X2 | . Then,
ϱ dG (y, X1 )dG (y, X2 ) + 4ν p2 |X1 | |X2 | + ν|Y |4p2 |X1 | |X2 |
|T | 6
y∈Y ′
+
β2 β2 + |X1 | |X2 |. p2 |X1 | p2 |X2 |
The next inequality is obtained by putting the following facts together: ν = α 2 ξ ϱ/64, β 6 γ p2 n3/2 , √ ′ n ′ γ 6 (α/2) ξ ϱ0 σ σ /24, q > α p, |X1 |, |X2 | > σ 2 and |Y | > σ n.
|T | 6 ϱ
(dG (y, X1 )dG (y, X2 )) +
y∈Y ′
ξ 4
ϱq2 |X1 | |X2 | |Y |.
(22)
√ ε ′ )q|Xi | for both i = 1, 2. Since (X , Y )G ∈ DISC(q, p, ε′ ), it is √ |X | |X | not hard to see that |Yi′′ | 6 ε ′ p 2 |Y |/(q|Xi |) for both i = 1, 2. Since |X1 |, |X2 | > ζ 2 , q > α p Define Yi′′ = y ∈ Y : dG (y, Xi ) > (1 +
and ε ′ 6 (α 3 ξ ϱ0 ζ /64)2 , the following holds for both i = 1, 2.
|Yi′′ | 6
ξ ϱα 2 64
|Y |.
Note that
(dG (y, X1 )dG (y, X2 )) =
(dG (y, X1 )dG (y, X2 ))
y∈Y ′ \(Y1′′ ∪Y2′′ )
y∈Y ′
+
(dG (y, X1 )dG (y, X2 ))
y∈Y ′ ∩(Y1′′ ∪Y2′′ )
6 |Y |(1 +
√ 2 2 ξ ϱα 2 |Y | 4p2 |X1 | |X2 | . ε ′ ) q |X1 | |X2 | + 32
√ Therefore, since ε ′ 6 ξ /24, the above inequality together with (22) implies ξ |T | 6 1 + ϱq2 |X1 | |X2 | |Y |, 2
a contradiction with (21).
4. Proof of the main result In this section we show how to combine the lemmas presented in Section 3 in order to prove Theorem 1.2.
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)
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Proof of Theorem 1.2. Let ε , α , η > 0, C > 1 and k > 4 be given. Let H1 , . . . , Hr be all the k-vertex 3-uniform hypergraphs which are linear and connector-free. Applying Lemma 3.1 with parameters k, C , ε and d = k for H1 , . . . , Hr , we obtain, respectively, constants δ1 , . . . , δr . Now put δmin = min{δ1 , . . . , δr }. Let δ ′ be given by Lemma 2.4 applied with α and δmin . Let ε ′ be given by Lemma 3.4 applied with α and δ ′ . Lemma 3.3 applied with ε ′ , η and σ = 1 gives δ . Following the quantification of Lemma 3.3 applied with α we obtain γ1 . Finally, following the quantification of Lemma 3.4 applied with σ = 1 we obtain γ2 . Put γ = min{γ1 , γ2 }. Let p = p(n) = o(1) with p ≫ n−2/(k−1) and let q = q(n) be such that α p 6 q 6 p. In what follows we suppose that n is sufficiently large. Let Γ = (V , EΓ ) be an n-vertex (p, β)-jumbled 3-uniform hypergraph and let G be a spanning n subhypergraph of Γ with |E (G)| = q 3 such that G satisfies Q′ (η, δ, q) and BDD(k, C , q). Suppose that β 6 γ p2 n3/2 . We want to prove that G contains (1 ± ε)nk q|E (H )| copies of all linear 3-uniform connector-free hypergraphs H with k vertices. By Lemma 3.3, our hypergraph G satisfies DISC(q, p, ε ′ ). Now apply Lemmas 2.4 and 3.4 in succession to deduce that G satisfies PAIR(q, p, δ ′ ) and TUPLE(δ, q). Now let H be any linear 3-uniform connector-free hypergraphs H with k vertices. Since G satisfies TUPLE(δ, q) and BDD(k, C , q), by Lemma 3.1, we conclude that
|Emb(H , G)| − nk q|E (H )| < εnk q|E (H )| . Acknowledgements We thank the referees for their helpful suggestions and comments. Y. Kohayakawa and G. O. Mota were partially supported by FAPESP (2013/03447-6) and CNPq (459335/2014-6). Y. Kohayakawa was also supported by FAPESP (2013/07699-0), CNPq (310974/2013-5), NUMEC/USP (Project MaCLinC/USP) and the NSF (DMS 1102086). G. O. Mota was also supported by FAPESP (2013/114312, 2013/20733-2). M. Schacht was supported by the Heisenberg-Programme of the DFG (grant SCHA 1263/4-1). A. Taraz was supported in part by DFG grant TA 309/2-2. This research was supported by a joint CAPES/DAAD PROBRAL project (Proj. 430/15). References [1] B. Bollobás, Extremal Graph Theory, in: London Mathematical Society Monographs, vol. 11, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978, MR506522. [2] F. Chung, R. Graham, Sparse quasi-random graphs, Combinatorica 22 (2) (2002) 217–244. http://dx.doi.org/10.1007/ s004930200010, Special issue: Paul Erdős and his mathematics, MR1909084 (2003d:05110). [3] F. Chung, R. Graham, Quasi-random graphs with given degree sequences, Random Struct. Algorithms 32 (1) (2008) 1–19. http://dx.doi.org/10.1002/rsa.20188, MR2371048 (2009a:05189). [4] F. Chung, R. Graham, R. Wilson, Quasi-random graphs, Combinatorica 9 (4) (1989) 345–362. http://dx.doi.org/10.1007/ BF02125347, MR1054011. [5] P. Erdős, Some old and new problems in various branches of combinatorics, in: Proc. 10th Southeastern Conference on Combinatorics, Graph Theory and Computing, 1979, pp. 19–37. [6] Y. Kohayakawa, G.O. Mota, M. Schacht, A. Taraz, Counting results for sparse pseudorandom hypergraphs I, European J. Combin. (2017) http://dx.doi.org/10.1016/j.ejc.2017.04.008, (in press). [7] Y. Kohayakawa, V. Rödl, Szemerédi’s regularity lemma and quasi-randomness, in: Recent Advances in Algorithms and Combinatorics, 2003, pp. 289–351. http://dx.doi.org/10.1007/0-387-22444-0_9, MR1952989 (2003j:05065). [8] Y. Kohayakawa, V. Rödl, M. Schacht, J. Skokan, On the triangle removal lemma for subgraphs of sparse pseudorandom graphs, in: An Irregular Mind, 2010, pp. 359–404. http://dx.doi.org/10.1007/978-3-642-14444-8_10, MR2815608. [9] M. Krivelevich, B. Sudakov, Pseudo-random graphs, in: More Sets, Graphs and Numbers, 2006, pp. 199–262. http://dx.doi. org/10.1007/978-3-540-32439-3_10, MR2223394 (2007a:05130). [10] J. Lenz, D. Mubayi, The poset of hypergraph quasirandomness, Random Struct. Algorithms 46 (4) (2015) 762–800. http:// dx.doi.org/10.1002/rsa.20521, MR3346466. [11] V. Nikiforov, On the edge distribution of a graph, Combin. Probab. Comput. 10 (6) (2001) 543–555. http://dx.doi.org/10. 1017/S0963548301004837, MR1869845. [12] Chr. Reiher, V. Rödl, M. Schacht, Embedding tetrahedra into quasirandom hypergraphs, ArXiv e-prints, Feb. 2016, Availble at arXiv:1602.02289. [13] V. Rödl, On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1–2) (1986) 125–134. MR837962 (88b:05098). [14] A. Thomason, Pseudorandom graphs, in: Random Graphs’85, (Poznań, 1985), 1987, pp. 307–331. MR89d:05158. [15] A. Thomason, Random graphs, strongly regular graphs and pseudorandom graphs, in: Surveys in Combinatorics 1987, (New Cross, 1987), 1987, pp. 173–195. MR88m:05072. [16] H. Towsner, σ -algebras for quasirandom hypergraphs, Random Struct. Algorithms 50 (1) (2017) 114–139. http://dx.doi. org/10.1002/rsa.20641.
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Counting results for sparse pseudorandom hypergraphs II Yoshiharu Kohayakawa a , Guilherme Oliveira Mota a , Mathias Schacht b , Anusch Taraz c a b c
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany Institut für Mathematik, Technische Universität Hamburg–Harburg, Hamburg, Germany
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abstract We present a variant of a universality result of Rödl (1986) for sparse, 3-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ‘‘pseudorandom’’ hypergraphs, which is proved in the companion paper (Counting results for sparse pseudorandom hypergraphs I). © 2017 Elsevier Ltd. All rights reserved.
1. Introduction We say that a graph G = (V , E ) satisfies property Q(η, δ, α) if, for every subgraph G[S ] induced by S ⊂ V with |S | > η|V |, we have
(α − δ)
|S | 2
|S | < E (G[S ]) < (α + δ) . 2
Rödl [13] (see also [7]) answered affirmatively a question posed by Erdős (see, e.g., [5,11] and [1, p. 363]) by proving the following result. Theorem 1.1. For all integers k > 1 and every α , η ∈ (0, 1), there exist δ > 0 and n0 such that the following holds for all integer n > n0 .
E-mail addresses: [email protected] (Y. Kohayakawa), [email protected] (G.O. Mota), [email protected] (M. Schacht), [email protected] (A. Taraz). http://dx.doi.org/10.1016/j.ejc.2017.04.007 0195-6698/© 2017 Elsevier Ltd. All rights reserved.
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Every n-vertex graph G that satisfies Q(η, δ, α) contains all graphs with k vertices as induced subgraphs. The quantification in Theorem 1.1 is what makes it unexpected. Indeed, note that η is not required to be small; it is allowed to be any constant less than 1. We establish a variant of Theorem 1.1 in the context of sparse 3-uniform hypergraphs. Our main result (see Theorem 1.2) allows one to count the number of copies (not necessarily induced) of certain fixed 3-uniform hypergraphs H in spanning subgraphs G of sparse jumbled 3-uniform hypergraphs Γ . Jumbled graphs are well studied due to the fruitful work of Thomason [14,15] and others (see also [2–4,9]). For 3-uniform hypergraphs we shall consider the following notion here. Let Γ = (V , E ) be a 3-uniform hypergraph and let X ⊂
V 2
and Y ⊂ V be given. Denote by EΓ (X , Y ) the set of pairs
({x1 , x2 }, y) ∈ X × Y such that {x1 , x2 , y} ∈ E and set eΓ (X , Y ) = |EΓ (X , Y )|. We say that Γ is (p, β)-jumbled if, for all subsets X ⊂
V 2
and Y ⊂ V , we have
eΓ (X , Y ) − p|X | |Y | 6 β |X | |Y |. A hypergraph H is called linear if every pair of edges shares at most one vertex. A hyperedge e of a linear 3-uniform hypergraph H is a connector if there exist a vertex v ̸∈ e and hyperedges e1 , e2 , and e3 each containing v and each intersecting e in exactly one vertex. Note that in the context of graphs the analogous definition would result in an edge e that intersects two edges e1 and e2 that a share vertex v outside e; in other words, the edges e, e1 , and e2 form a graph triangle. Our result can be viewed as a counting lemma for fixed linear, connector-free 3-uniform hypergraphs H contained in certain spanning 3-uniform subhypergraphs Gn of (p, o(p2 n3/2 ))-jumbled hypergraphs Γ for sufficiently large p and n. In comparison to Theorem 1.1 one may think of Γ in our result as replacing the complete graph Kn , which contains the spanning graph G satisfying a ‘‘certain’’ condition Q(η, δ, α) on the distribution of its edge set. Below we define the variant of Q(η, δ, α) for 3-uniform hypergraphs that is imposed on Gn . We say that a 3-uniform hypergraph G = (V , E ) satisfies property Q′ (η, δ, q) if, for all X ⊂
and Y ⊂ V with |X | > η
|V | 2
V 2
and |Y | > η|V |, we have
(1 − δ)q|X | |Y | 6 EG (X , Y ) 6 (1 + δ)q|X | |Y |. Considering the cardinality of EG (X , Y ) for certain X ⊂
(1)
V 2
and Y ⊂ V to obtain information on
the subhypergraphs of G has recently been shown to be fruitful (see [12]). Since we consider sparse 3-uniform hypergraphs here, we also require a boundedness condition. Given a pair {v1 , v2 } ∈
V 2
,
we define NG ({v1 , v2 }) = v3 ∈ V : {v1 , v2 , v3 } ∈ E
and we say that G = (V , E ) satisfies BDD(k, C , q) if, for all 1 6 r 6 k and for all distinct 2-element
sets S1 , . . . , Sr ∈
V 2
, we have
|NG (S1 ) ∩ · · · ∩ NG (Sr )| 6 Cnqr . Finally, we can state the variant of Theorem 1.1 for 3-uniform hypergraphs considered here. In what follows, given hypergraphs H and G, we let Emb(H , G) be the set of embeddings of H into G, i.e., the set of edge preserving injective maps ϕ : V (H ) → V (G). Theorem 1.2. For all constants ε , α , η ∈ (0, 1) and C > 1, and every integer k > 5, there exist δ , γ > 0 such that if p = p(n) ≫ n−2/(k−1) and p = p(n) = o(1) and n is sufficiently large, then the following holds for every α p 6 q 6 p and every β 6 γ p2 n3/2 . Suppose that
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(i) Γ = (V , EΓ ) is an n-vertex (p, β)-jumbled 3-uniform hypergraph and n (ii) G = (V , EG ) is a spanning subhypergraph of Γ with |EG | = q 3 and G satisfies Q′ (η, δ, q) and BDD(k, C , q). Then for every linear 3-uniform connector-free hypergraph H on k vertices we have
|Emb(H , G)| − nk q|E (H )| < εnk q|E (H )| . We close this introduction with a few remarks concerning Theorem 1.2. Note that if p ≫ n−1/4 , then the random 3-uniform hypergraph, where each possible edge exists with probability p independently of all other edges, is (p, γ p2 n3/2 )-jumbled with high probability for all γ > 0. Therefore, our result applies to appropriate spanning subhypergraphs of such random 3-uniform hypergraphs. The hypergraph H in Theorem 1.2 is required to be linear and connector-free. It is possible that both conditions could be further relaxed. In fact, in the context of dense hypergraphs the assumption imposed on the edge distribution of G from (1) yields the right number of embeddings for a slightly larger class (see, e.g., [10,16] for details). Also it might be possible that the condition that H should be connector-free is not necessary. We inherit this condition from a counting lemma proved in [6] (see Lemma 3.1) and this is the only reason why we impose it here. Most of the definitions generalise naturally to r-uniform hypergraphs for r larger than 3, and it would be interesting to extend Theorem 1.2 to r-uniform hypergraphs with r > 3. Unfortunately such a generalisation presents new difficulties and we leave this open here. Organisation. We give an overview of the proof of Theorem 1.2 in Section 2. The proof will be based on some intermediate lemmas, which we establish in Section 3. In Section 4 we present the proof of Theorem 1.2, which, after all the work in Section 3, will be quite short. 2. Overview of the proof The first step in the proof of Theorem 1.2 is to show that, under the conditions of the theorem, G satisfies a strong property involving degrees and co-degrees (see Lemmas 2.4, 3.3 and 3.4). After that we apply a counting result (Lemma 3.1). Before we discuss the scheme of the proof, let us define some hypergraph properties, called Discrepancy, Pair, and Tuple. Definition 2.1 (DISC—Discrepancy Property). Let G = (V , E ) be a 3-uniform hypergraph and let X , Y ⊂ V be given. We say that the pair (X , Y ) satisfies DISC(q, p, ε) in (G or (X , Y )G satisfies DISC(q, p, ε)) if for all X ′ ⊂
X 2
and Y ′ ⊂ Y we have
eG (X ′ , Y ′ ) − q|X ′ | |Y ′ | 6 εp |X | |Y |. 2
Furthermore, if (V , V ) satisfies DISC(q, p, ε) in G, then we say that the hypergraph G satisfies DISC(q, p, ε). For a 3-uniform hypergraph G = (V , E ), a set of vertices Y ⊂ V , and pairs S1 , S2 ∈
V 2
we denote
NG (S1 ) ∩ Y by NG (S1 ; Y ) and NG (S1 ) ∩ NG (S2 ) ∩ Y by NG (S1 , S2 ; Y ). Definition 2.2 (PAIR—Pair Property). Let G = (V , E ) be a 3-uniform hypergraph and let X , Y ⊂ V be given. We say that the pair (X , Y ) satisfies PAIR(q, p, δ) in (G or (X , Y )G satisfies PAIR(q, p, δ)) if the following conditions hold:
|NG (S1 ; Y )| − q|Y | 6 δ p |X | |Y |, X
2
S1 ∈ 2
X
S1 ∈ 2
2 |NG (S1 , S2 ; Y )| − q2 |Y | 6 δ p2 |X | |Y |. X
S2 ∈ 2
2
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Furthermore, if (V , V ) satisfies PAIR(q, p, δ) in G, then we say that the hypergraph G satisfies PAIR(q, p, δ). Definition 2.3 (TUPLE—Tuple Property). We define TUPLE(δ, q) as the family of n-vertex 3-uniform hypergraphs G = (V , E ) such that the following two conditions hold: (i) |NG (S1 )| − nq < δ nq for all but at most δ
n 2
sets S1 ∈
V 2
;
n V (ii) |NG (S1 ) ∩ NG (S2 )| − nq2 < δ nq2 for all but at most δ ( 22 ) pairs {S1 , S2 } of distinct sets in 2 . The next result allows us to obtain property TUPLE from PAIR. Since its proof is simple we will omit it. Lemma 2.4. For all 0 < α 6 1 and 0 < δ < 1 there exists δ ′ > 0 such that if a 3-uniform hypergraph G satisfies PAIR(q, p, δ ′ ) for α p 6 q 6 p, then G satisfies TUPLE(δ, q). In what follows we explain the organisation of the proof. Consider the setup of Theorem 1.2. In order to obtain the conclusion of the theorem, we will use a counting result (Lemma 3.1), which requires that G satisfies properties BDD and TUPLE for the appropriate parameters. Since G satisfies BDD by hypothesis, it suffices to prove that G satisfies TUPLE. Using Lemma 3.3 it is possible to obtain DISC from property Q′ . Then, using that G satisfies DISC one can show that G satisfies PAIR using Lemma 3.4, which implies TUPLE by Lemma 2.4. The quantification used in these implications is carefully analysed in Section 4. 3. Main lemmas We start by stating the counting lemma needed in the proof of Theorem 1.2. In order to apply it to a 3-uniform n-vertex hypergraph G, we shall prove that G satisfies TUPLE(δ, q) for a sufficiently small δ and sufficiently large 0 < q = q(n) 6 1. Since Lemma 2.4 allows us to obtain TUPLE from PAIR, we need to proof that G satisfies PAIR(q, p, δ ′ ) for a sufficiently small δ ′ and appropriate functions p and q. This is done using Lemmas 3.3 and 3.4, which are proved, respectively, in the Sections 3.1 and 3.2. Given a 3-uniform hypergraph H, we define parameters dH = max{δ(J ) : J ⊂ H } and DH = min{3dH , ∆(H )}, where δ(J ) and ∆(J ) stand, respectively, for the minimum and maximum degree of a vertex in V (J ). The following result, proved in [6], is our counting lemma. Lemma 3.1. Let k > 5 be an integer and let ε > 0, C > 1 and an integer d > 2 be fixed. Let H be a linear 3-uniform connector-free hypergraph on k vertices such that DH 6 d. Then, there exists δ > 0 for which the following holds for any q = q(n) with q ≫ n−1/d and q = o(1 )n and for sufficiently large n. If G is an n-vertex 3-uniform hypergraph with |E (G)| = q 3 hyperedges and G satisfies BDD(DH , C , q) and TUPLE(δ, q), then
|Emb(H , G)| − nk q|E (H )| < εnk q|E (H )| . Recall that p ≫ n−2/(k−1) is one of the conditions in Theorem 1.2. Under this condition on p, since DH 6 (k − 1)/2 for any linear 3-uniform hypergraph H on k vertices, we can apply Lemma 3.1 with any q = q(n) such that q > α p for some 0 6 α 6 1. 3.1. Q′ implies DISC Given a 3-uniform hypergraph G = (V , E ) and subsets A ⊂
V 2
and non-empty B ⊂ V , the
q-density between A and B is defined as dq (A, B) =
|EG (A, B)| . q|A| |B|
Before we state the main result of this subsection, Lemma 3.3, we shall prove the following result, which is inspired by a result in [13] for graphs.
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Lemma 3.2. For all 0 < η < 1 and 0 < ε ∗ < (1 − η)/3, there exists δ > 0 such that, if G = (V , E ) is ′ an n-vertex 3-uniform hypergraph that satisfies Q (η, δ, q), then the following holds. For every C ⊂
V 2
and D ⊂ V such that |C | is a multiple of ⌈ε ∗
n ⌉ and |D| is a multiple of ⌈ε ∗ n⌉, 2
we have 1 − ε ∗ < dq (C , D) < 1 + ε ∗ . Proof. Fix η > 0 and 0 < ε ∗ < (1 − η)/3. Let δ = ε ∗ 3 /24 and put t = ⌈1/ε ∗ ⌉. Suppose G = (V , E ) is an n-vertex 3-uniform hypergraph that satisfies Q′ (η, δ, q). Now, fix C ⊂
V 2
and D ⊂ V such
⌉ and |D| = k2 ⌈ε n⌉ for some positive integers k1 and k2 . Let C1 , . . . , Ck1 and n ⌉ and |D1 | = 2 V ∗ · · · = |Dk2 | = ⌈ε n⌉. Now we partition the sets 2 \ C and V \ D, respectively, in sets Ck1 +1 , . . . , Ct n and Dk2 +1 , . . . , Dt such that |Ck1 +1 | = · · · = |Ct −1 | = ⌈ε ∗ 2 ⌉ and |Dk2 +1 | = · · · = |Dt −1 | = ⌈ε ∗ n⌉. n Note that |Ct | 6 ε ∗ 2 and |Dt | 6 ε ∗ n. We divide the rest of the into two parts. First, we prove that for any triple i, j, j′ ∈ [t − proof 1], |e(Ci , Dj ) − e(Ci , Dj′ )| 6 6δ 2n nq, and for any triple i, i′ , j ∈ [t − 1], |e(Ci , Dj ) − e(Ci′ , Dj )| 6 6δ 2n nq. To finish the proof we put these estimates together to show that 1 − ε ∗ < dq (C , D) < 1 + ε ∗ . ∗ Put X = C2 ∪ · · · ∪ Ct and Y = D3 ∪ · · · ∪ Dt . Since ε < (1 − η)/3, we have |X | = (t − 2)⌈ε ∗ 2n ⌉ + |Ct | > (t − 2)ε∗ 2n > η 2n and |Y | = (t − 3)⌈ε ∗ n⌉ + |Dt | > (t − 3)ε∗ n > ηn. Therefore, using Q′ (η, δ, q) and |D1 | = |D2 |, the following two inequalities hold. e(X , D1 ∪ Y ) − e(X , D2 ∪ Y ) 6 2δ|X |(|D1 | + |Y |)q, (2) e(C1 ∪ X , Y ) e(C1 ∪ X , Dj ∪ Y ) (3) (|C | + |X |)|Y |q − (|C | + |X |)(|D | + |Y |)q 6 2δ, for j ∈ {1, 2}. 1 1 j that |C | = k1 ⌈ε
∗ n
∗
2
D1 , . . . , Dk2 be, respectively, partitions of C and D such that |C1 | = · · · = |Ck1 | = ⌈ε ∗
Now we define the following for j ∈ {1, 2} p1j =
e(C1 ∪ X , Y ) + e(X , Dj ) e(C1 ∪ X , Y ) − . (|C1 | + |X |)|Y |q (|C1 | + |X |)(|Dj | + |Y |)q
By (3), the following holds for j ∈ {1, 2} e(C1 , Dj ) 6 p1j + 2δ. (4) (|C1 | + |X |)(|Dj | + |Y |)q Note that e(X , D1 )− e(X , D2 ) = e(X , D1 ∪ Y )− e(X , D2 ∪ Y ). Thus, using (2), we obtain the following p1j − 2δ 6
inequality.
|p11 − p12 | =
e(X , D1 ) − e(X , D2 )
|X | 6 2δ < 2δ. (|C1 | + |X |)(|D1 | + |Y |)q |C1 | + |X |
(5)
Putting (4) and (5) together, we obtain the following inequality.
|e(C1 , D1 ) − e(C1 , D2 )| < 6δ(|C1 | + |X |)(|D1 | + |Y |)q < 6δ
n 2
nq.
Applying the same strategy one can prove that, for any triple i, j, j′ ∈ [t − 1],
|e(Ci , Dj ) − e(Ci , Dj′ )| < 6δ
n 2
nq.
(6)
Analogously, we obtain the following equation for any triple i, i′ , j ∈ [t − 1].
|e(Ci , Dj ) − e(Ci′ , Dj )| < 6δ
n 2
nq.
(7)
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By (6) and (7), we have |e(Ci , Dj ) − e(Ci′ , Dj′ )| < 12δ
|dq (Ci , Dj ) − dq (Ci′ , Dj′ )| <
12δ
n 2
nq
|Ci | |Dj |q
<
n
12δ
(ε∗ )2
2
=
)
–
nq for any i, i′ , j, j′ ∈ [t − 1]. Therefore,
ε∗
(8)
2
holds for any i, i′ , j, j′ ∈ [t − 1]. Put WC = C1 ∪· · ·∪ Ct −1 and WD = D1 ∪· · ·∪ Dt −1 . Since |WC | > η and |WD | > ηn, we know, by property Q′ (η, δ, q), that 1 − δ < dq (WC , WD ) < 1 + δ.
n 2
(9)
Suppose for a contradiction that there exist indexes i0 , j0 ∈ [t − 1] such that either dq (Ci0 , Dj0 ) > 1 + ε ∗ or dq (Ci0 , Dj0 ) < 1 − ε ∗ . Then, by (8), either for all i, j ∈ [t − 1] we have dq (Ci , Dj ) > 1 + ε ∗ /2 or for all i, j ∈ [t − 1] we have dq (Ci , Dj ) < 1 − ε ∗ /2. But note that
dq (WC , WD ) =
i,j∈[t −1]
dq (Ci , Dj )|Ci | |Dj |q
.
|WC | |WD |q
Then, either
(t − 1)2 (1 − ε ∗ /2) ε ∗ 2n ⌈ε ∗ n⌉q = (1 − ε ∗ /2) < 1 − δ, dq (WC , WD ) < |WC | |WD |q or
(t − 1)2 (1 + ε ∗ /2) ε ∗ 2n ⌈ε ∗ n⌉q dq (WC , WD ) > = (1 + ε ∗ /2) > 1 + δ, |WC | |WD |q a contradiction with (9). Therefore, for all i, j ∈ [t − 1], 1 − ε ∗ < dq (Ci , Dj ) < 1 + ε ∗ .
(10)
It remains to estimate the densities dq (Ck1 , Dj ) and dq (Ci , Dk2 ) with k1 = t and k2 = t for all 1 6 i 6 k1 n n and 1 6 j 6 k2 . Note that k1 = t and k2 = t imply, respectively, ⌈ε ∗ 2 ⌉ = ε ∗ 2 and ⌈ε ∗ n⌉ = ε ∗ n, but in these cases one can prove in the same way we proved (10). Therefore, putting all these estimates together, we obtain 1 − ε ∗ < dq (C , D) < 1 + ε ∗ . The next lemma shows how one can obtain discrepancy properties from Q′ in spanning subhypergraphs of sufficiently jumbled 3-uniform hypergraphs. Lemma 3.3. For all 0 < ε ′ , η, σ < 1 there exists δ > 0 such that for every α > 0 there exists γ > 0 such that the following holds. Let Γ = (V , EΓ ) be an n-vertex (p, β)-jumbled 3-uniform hypergraph for 0 < p = p(n) 6 1 such that α p 6 q 6 p and β 6 γ pn3/2 . Let G = (V , EG ) be a spanning subhypergraph of Γ . If G satisfies Q′ (η, δ, q), then every pair (X , Y )G with X , Y ⊂ V such that |X |, |Y | > σ n satisfies DISC(q, p, ε ′ ). Proof. Fix ε ′ , η, σ > 0 and let ε ∗ = min ε ′ σ 2 /24, (1 − η)/4 . Let δ ′ be the constant given by Lemma 3.2 applied with η and ε ∗ . Put δ = min{δ ′ , ε ′ }, α > 0 and γ = σ 3/2 αε ′ /2. Suppose that α p 6 q 6 p and β 6 pγ n3/2 . Let Γ = (V , EΓ ) be an n-vertex (p, β)-jumbled 3-uniform hypergraph and let G = (V , EG ) be a spanning subhypergraph of Γ such that G satisfies Q′ (η, δ, q). Let (X , Y )G be a pair with X , Y ⊂ V such that |X |,|Y | > σ n. We want to prove that
2
(X , Y )G satisfies DISC(q, p, ε′ ). For this, fix arbitrary subsets X ′ ⊂ X2 and Y ′ ⊂ Y . We will prove that eG (X ′ , Y ′ ) − q|X ′ | |Y ′ | 6 ε ′ p |X | |Y |. 2 |X | Upper bound. First, consider the case where |X ′ | 6 ε ′ 2 or |Y ′ | 6 ε ′ |Y |. Note that, from the choice of γ and β , since |X |, |Y | > σ n, we have |X | |Y |. (11) β |X ′ | |Y ′ | 6 αε ′ p 2
Y. Kohayakawa et al. / European Journal of Combinatorics (
)
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Therefore, eG (X ′ , Y ′ ) 6 p|X ′ | |Y ′ | + β
|X ′ | |Y ′ |
6 q|X ′ | |Y ′ | + (1 − α)p|X ′ | |Y ′ | + β
|X ′ | |Y ′ | |X | 6 q|X ′ | |Y ′ | + (1 − α)pε ′ |Y | + β |X ′ | |Y ′ | 2 | X | 6 q|X ′ | |Y ′ | + ε ′ p |Y |,
(12)
2
where the first inequality follows from the jumbledness of Γ and the fact that G is a subhypergraph of
Γ , the second one follows from the value of q, the third one follows from the fact that |X ′ | 6 ε ′ |X2 | |X | or |Y ′ | 6 ε ′ |Y |, and the last one is a consequence of (11). Thus, we may assume |X ′ | > ε ′ 2 and
|Y ′ | > ε′ |Y |. We consider four the size of |X ′ | and |Y ′ |. cases, ′depending on n n ′ ∗ ∗ Case 1: (|X | > (1 − ε ) 2 and |Y | > (1 − ε )n). By the choice of ε ∗ , we have |X ′ | > η 2 and ′ ′ |Y | > ηn. By Q (η, δ, q) we conclude that |X | ′ ′ ′ ′ ′ ′ ′ eG (X , Y ) 6 (1 + δ)q|X | |Y | 6 q|X | |Y | + ε p |Y |. 2
Case 2: (|X ′ | < (1 − ε ∗ )
n 2
and |Y ′ | < (1 − ε ∗ )n). Note that, since |X ′ | < (1 − ε ∗ )
|Y | < (1 − ε )n, there exist subsets X ⊂ ′
∗
V 2
∗
∗
∗
′
′′
∗
n 2
and
′
and Y ⊂ V such that X = X ∪ X and Y = Y ∪ Y ′′ ,
n n with X ∩ X = ∅ and Y ∩ Y = ∅, where |X | 6 ε ∗ 2 and |X ∗ | is multiple of ⌈ε ∗ 2 ⌉, and |Y ′′ | 6 ε ∗ n ∗ ∗ and |Y | is a multiple of ⌈ε n⌉. Then, we can use Lemma 3.2 to obtain the following inequality.
′
′′
′
′′
′′
eG (X ′ , Y ′ ) 6 eG (X ∗ , Y ∗ ) 6 (1 + ε ∗ )|X ∗ | |Y ∗ |q 6 (1 + ε ∗ )q|X ′ | |Y ′ | + 2q |X ′ | |Y ′′ | + |X ′′ | |Y ′ | + |X ′′ | |Y ′′ | .
Since ε ∗ 6 ε ′2 σ 2 /16, we have |X ′′ | 6 ε ∗
n 2
6 (ε ′ /8)|X ′ | and |Y ′′ | 6 ε ∗ n 6 (ε ′ /8)|Y ′ |. Therefore,
eG (X ′ , Y ′ ) 6 (1 + ε ∗ )q|X ′ | |Y ′ | + 2q 3(ε ′ /8)|X ′ | |Y ′ |
6 q|X ′ | |Y ′ | +
ε′ 4
q|X ′ | |Y ′ | +
3ε ′ 4
q|X ′ | |Y ′ | 6 q|X ′ | |Y ′ | + ε ′ p
|X |
2
|Y |.
Case 3: (|X ′ | > (1 − ε ∗ ) 2 and |Y ′ | < (1 − ε ∗ )n). As noticed before, since |Y ′ | < (1 − ε ∗ )n, there exist subsets Y ∗ , Y ′′ ⊂ V such that Y ∗ = Y ′ ∪ Y ′′ with Y′ ∩ Y ′′ = ∅, where |Y ′′ | 6 ε ∗ n and |Y ∗ | is a
n
multiple of ⌈ε ∗ n⌉. Note that there exist subsets X˜ , X ′′ ⊂
V such that X ′ = X˜ ∪ X ′′ with X˜ ∩ X ′′ = ∅, 2 n where |X ′′ | 6 ε 2 and |X˜ | is a multiple of ⌈ε ∗ 2 ⌉. ′′ ′′ ′′ then we ‘‘complete’’ X ′′ with elements of If X is empty, then put W = ∅. If X is not empty, V ′′ ′′ ′′ ′′ ∗ n to obtain W such that X ⊂ W and |W | = ⌈ε 2 ⌉ (note that possibly W ′′ ∩ X˜ ̸= ∅). Thus, 2 |X˜ | + |W ′′ | 6 |X ′ | + ε ∗ 2n . By using Lemma 3.2, we have
∗ n
eG (X ′ , Y ′ ) 6 eG (W ′′ , Y ∗ ) + eG (X˜ , Y ∗ )
6 (1 + ε ∗ )q |Y ∗ | |W ′′ | + |Y ∗ | |X˜ |
= (1 + ε ∗ )q |Y ′ | |W ′′ | + |Y ′′ | |W ′′ | + |Y ′ | |X˜ | + |Y ′′ | |X˜ | n n 6 (1 + ε ∗ )q |Y ′ | |X ′ | + ε ∗ + |Y ′′ | |X ′ | + ε ∗ 2n 2 ∗ ′ ′ ∗ ∗ n ∗ ′ ′ 6 (1 + ε )q|X | |Y | + 2q ε |Y | + |X |ε n + ε ε∗ n . 2
2
8
Y. Kohayakawa et al. / European Journal of Combinatorics (
Since ε ∗ 6 ε ′2 σ 2 /16, we have ε eG (X ′ , Y ′ ) 6 q|X ′ | |Y ′ | +
∗ n
ε′ 4
6 q|X ′ | |Y ′ | + ε ′
Case 4: (|X ′ | < (1 − ε ∗ )
n
Lower bound. If |X ′ | 6 ε ′
2
2
|X |
2
X˜ ⊂
3ε ′ 8
|X ′ | |Y ′ |
|Y |p.
and |Y ′ | > (1 − ε ∗ )n). This case is analogous to Case 3.
|X |
2
or |Y ′ | 6 ε ′ |Y |, then there is nothing to prove, because ε ′
q|X | |Y |. Therefore, assume that |X | > ε ′
–
6 (ε ′ /8)|X ′ | and ε ∗ n 6 (ε ′ /8)|Y ′ |. Therefore,
q|X ′ | |Y ′ | + 2q
)
′
′
′
|X | 2
|X | 2
|Y |p >
and |Y | > ε |Y |. Clearly, there exist subsets ′
′
V 2
, Y˜ ⊂ V such that X ′ = X˜ ∪ X ′′ , Y ′ = Y˜ ∪ Y ′′ , with X˜ ∩ X ′′ = Y˜ ∩ Y ′′ = ∅, where |X ′′ | 6 ε ∗ 2n , |X˜ | is a multiple of ⌈ε ∗ 2n ⌉, |Y ′′ | 6 ε ∗ n and |Y˜ | is a multiple of ⌈ε ∗ n⌉. Since ε ∗ 6 ε ′2 σ 2 /8, we have n |X ′′ | 6 ε∗ 6 (ε ′ /4)|X ′ | 6 ε ′ /4(1 − ε ∗ ) |X ′ | 2
and |Y | 6 ε n 6 (ε ′ /4(1 − ε ∗ ))|Y ′ |. Then, by Lemma 3.2, since eG (X ′ , Y ′ ) > eG (X˜ , Y˜ ), we have ′′
∗
eG (X ′ , Y ′ ) > (1 − ε ∗ )|X˜ | |Y˜ |q
= (1 − ε ∗ )q |X ′ | |Y ′ | − |X ′ | |Y ′′ | − |X ′′ | |Y ′ | + |X ′′ | |Y ′′ | > (1 − ε ∗ )q|X ′ | |Y ′ | − (1 − ε ∗ )q |X ′ | |Y ′′ | + |X ′′ | |Y ′ | > q|X ′ | |Y ′ | − ε ∗ q|X ′ | |Y ′ | − (1 − ε ∗ )q (ε ′ /2(1 − ε ∗ ))|X ′ | |Y ′ | > q|X ′ | |Y ′ | −
ε′ 2
> q|X ′ | |Y ′ | − ε ′
q|X ′ | |Y ′ | −
|X | 2
ε′ 2
q|X ′ | |Y ′ |
|Y |p.
3.2. DISC implies PAIR The next lemma, which is a variation of Lemma 9 in [8], makes it possible to obtain PAIR from DISC in spanning subhypergraphs of sufficiently jumbled 3-uniform hypergraphs. Lemma 3.4. For all 0 < α 6 1 and δ ′ > 0 there exists ε ′ > 0 such that for all σ > 0 there exist γ > 0 such that the following holds for sufficiently large n. Suppose that
√
(i) Γ = (V , EΓ ) is an n-vertex 3-uniform (p, β)-jumbled hypergraph with p > 1/ n, (ii) G = (V , EG ) is a spanning subhypergraph of Γ , and (iii) X , Y ⊂ V with |X |, |Y | > σ n. Then, the following holds. If β 6 γ p2 n3/2 and (X , Y )G satisfies DISC(q, p, ε ′ ) for some q with α p 6 q 6 p, then (X , Y )G satisfies PAIR(q, p, δ ′ ). We need the following results in order to prove Lemma 3.4. First, consider the following fact, which is similar to [8, Fact 13]. Fact 3.5. Let Γ be a 3-uniform (p, β)-jumbled hypergraph. Let U ⊂
V 2
and W ⊂ V and ξ > 0. If we
have |NΓ ({x, y}, W )| > (1 + ξ )p|W | for every {x, y} ∈ U or we have |NΓ ({x, y}, W )| 6 (1 − ξ )p|W |
Y. Kohayakawa et al. / European Journal of Combinatorics (
)
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9
for every {x, y} ∈ U, then
|U | |W | 6
β2 . ξ 2 p2
Proof. Let Γ , U, W and ξ be as in the statement and suppose that for every {x, y} ∈ U we have NΓ ({x, y}, W ) > (1 + ξ )p|W |. Suppose for a contradiction that |U | |W | > β2 22 . Then, eΓ (U , W ) > ξ p √ |U |(1 + ξ )p|W | > p|U | |W | + β |U | |W |, a contradiction to the jumbledness of Γ . The case where |NΓ ({x, y}, W )| 6 (1 − ξ )p|W | for every {x, y} ∈ U is analogous. Our next result, Lemma 3.8, is very similar to [8, Lemma 21], but in Lemma 3.8 we consider bipartite
, V ; EΓ instead of Γ = (U , V ; EΓ ) in [8], and we consider subsets X1 , X2 of V2 n with |X1 |, |X2 | > η 2 instead of subsets X1 , X2 of V with |X1 |, |X2 | > ηn. Due to this fact, the value of β in Lemma 3.8 is γ p2 n3/2 , while in [8, Lemma 21] we have β = γ pn. The proof of Lemma 3.8 is
graphs Γ =
V 2
identical to the proof of [8, Lemma 21] and we omit it here. Let Γ = (V , EΓ ) be a graph and let X , Y ⊂ V . As usual, we denote by eΓ (X , Y ) the number of edges of Γ with one end-vertex in X and one end-vertex in Y , where edges contained in X ∩ Y are counted twice. We need to define jumbledness and discrepancy for graphs. Definition 3.6 (Jumbledness for Graphs). We say that Γ = √(V , EΓ ) is a (p, β)-jumbled graph if, for all subsets X , Y ⊂ V , we have eΓ (X , Y ) − p|X | |Y | 6 β |X | |Y |. Furthermore, a bipartite graph
ΓB = (U , V ; E ) is called (p, β)-jumbled if, for all X ⊂ U and Y ⊂ V , we have eΓ (X , Y ) − p|X | |Y | 6 √ β |X | |Y |.
Definition 3.7 (Discrepancy for Graphs). Let G = (V , E ) be a graph and let X , Y ⊂ V be disjoint. We say that (X , Y ) satisfies DISC(q, p, ε) in G (or (X , Y )G satisfies DISC(q, p, ε)) if for all X ′ ⊂ X and Y ′ ⊂ Y we have
eG (X ′ , Y ′ ) − q|X ′ | |Y ′ | 6 ε p|X | |Y |. Lemma 3.8. For all positive real ϱ0 and ν , there exists a positive real µ such that, for all σ ′ > 0, there exist γ > 0 and n0 > 0 such that for all n > n0 , the following holds. Suppose
√ , V ; EΓ is a bipartite (p, β)-jumbled graph with |V | > n, p > 1/ n and β 6 γ p2 n3/2 , n V (ii) X1 , X2 ⊂ 2 and Y ⊂ V with |X1 |, |X2 | > σ ′ 2 , |Y | > σ ′ n, (iii) B = (X1 , X2 ; EB ) is an arbitrary bipartite graph. (i) Γ =
V 2
Then, if (X1 , X2 )B satisfies DISC(ϱ, 1, µ) for some ϱ with ϱ0 6 ϱ 6 1, then for all but at most ν|Y | vertices y ∈ Y , the pair NΓ (y, X1 ), NΓ (y, X2 ) B satisfies DISC(ϱ, 1, ν). We need two more facts before proving of Lemma 3.4. Fact 3.9 ([8, Fact 22]). Suppose ϱ0 > 0, µ > 0 and B = (X , EB ) is a graph with |EB | > ϱ0 there exist disjoint subsets X1 , X2 ⊂ X such that
|X | 2
. Then
(i) (X1 , X2 )B satisfies DISC(ϱ, 1, µ) for some ϱ > ϱ0 ,
(ii) |X1 |, |X2 | > ζ |X | for ζ = ϱ0 100/µ /4. 2
Fact 3.10. Let Γ = (V , E ) be a 3-uniform hypergraph and let Γ ′ = where E ′ = {{v1 , v2 }, v} : {v1 , v2 } ∈
and only if Γ ′ is (p, β)-jumbled.
V 2
V 2
, V ; E ′ be a bipartite graph,
, v ∈ V and {v1 , v2 , v} ∈ E . Then, Γ is (p, β)-jumbled if
10
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)
–
We have stated all the tools needed in the proof of Lemma 3.4. This proof is very similar to the proof of [8, Lemma 9]. Proof of Lemma 3.4. Let 0 < α 6 1 and 0 < δ ′ < 1 be given. Put ξ = δ ′ /6, ϱ0 = δ ′ /50 and ν = α 2 ξ ϱ0 /64. Let µ be obtained by an application of Lemma 3.8 with parameters ϱ0 and ν . Without 100/µ2 loss of generality, assume µ < ξ ϱ0 /4. Let ζ = ϱ0 /4 be given and put ε ′ = min αδ ′2 /36, (α 3 ξ ϱ0 ζ /64)2 . Now fix σ > 0 and let σ ′ = ζ σ 2 /2. Following the quantification of Lemma 3.8 applied with parameter σ ′ we obtain γ ′ and n0 . Then, put γ = min γ ′ , σ 3 δ ′ /12, (α/2) ξ ϱ0 σ σ ′ /24 . √ Finally, consider n sufficiently large and suppose p > 1/ n. 2 3/2 Fix β 6 γ p n and consider a 3-uniform (p, β)-jumbled hypergraph Γ = (V , EΓ ) such that |V | = n and let G = (V , EG ) be a spanning subhypergraph of Γ . Let X , Y be subsets of V such that ′ |X |, |Y| > σ n. Suppose that (X , Y )G satisfies DISC(q, p, ε ) for some q with α p 6 q 6 p, i.e., for all X X ′ ⊂ 2 and Y ′ ⊂ Y the following holds.
eG (X ′ , Y ′ ) − q|X ′ | |Y ′ | 6 ε ′ p |X | |Y |.
(13)
2
We want to prove that the following inequalities hold:
|NG (S1 ; Y )| − q|Y | 6 δ ′ p |X | |Y |,
(14)
2
X
S1 ∈ 2
X
S1 ∈ 2
2 |NG (S1 , S2 ; Y )| − q2 |Y | 6 δ ′ p2 |X | |Y |.
(15)
2
X
S2 ∈ 2
We start by verifying (14). For at most δ ′
|X |
2
/6 pairs S ∈
X 2
, we have
|NG (S , Y )| − q|Y | > (δ ′ /3)q|Y |.
with at least δ ′
/12 elements such that, for all {x, x } ∈ BX , either |NG ({x, x }, Y )| > (1 + δ /3)q|Y | or for all of them we have |NG ({x, x′ }, Y )| < (1 − δ ′ /3)q|Y |. In either case, we would have ′2 ′2 eG (BX , Y ) − q|BX | |Y | > δ q |X | |Y | > δ α p |X | |Y | > ε ′ p |X | |Y |, Indeed, otherwise there would be a set BX ⊂ ′
′
X 2
|X | 2
′
36
36
2
2
2
where the last inequality follows from the choice of ε . But this contradicts (13) when we put X ′ = BX and Y ′ = Y . X Let U be the set of pairs S ∈ 2 such that |NΓ (S , Y )| > 2p|Y |. By Fact 3.5 applied with U and Y ′
with ξ = 1, we know that there exist at most β 2 /p2 |Y | elements S ∈ U such that |NΓ (S , Y )| > 2p|Y |. Therefore,
′ ′ |NG (S , Y )| − q|Y | 6 |X | δ q|Y | + δ |X | 2p|Y | + (β 2 /p2 |Y |)|Y | 2
X
3
6
2
S∈ 2
6p
|X | 2
|Y |
2δ ′ 3
+ (β/p)2 6 δ ′ p
|X |
2
where the last inequality follows from the facts that β 6 γ p2 n3/2 and γ 6 that (14) holds.
|Y |, σ 3 δ ′ /12. We just proved
Y. Kohayakawa et al. / European Journal of Combinatorics (
)
–
11
Suppose for a contradiction that (15) does not hold. Then,
2 |NG (S1 , S2 ; Y )| − q2 |Y | > δ ′ p2 |X | |Y |.
X
S1 ∈ 2
(16)
2
X
S2 ∈ 2
Define the following sets of ‘‘bad’’ pairs.
B1 = (S1 , S2 ) ∈ B2 = (S1 , S2 ) ∈
X
2 X
2
X
×
2
×
X
2
: |NΓ (S1 , Y )| > 2p|Y | , \ B1 : |NΓ (S1 , S2 , Y )| > 4p2 |Y | .
Since Γ is (p, β)-jumbled, it follows that
|B1 | 6
β2 2 p |Y |
|X | 2
6
γ 2 n3 p2 |Y |
|X |
6
2
γ 2 n2 p 2 σ
|X |
6
2
δ′ 3
p2
|X |
2
2
where the first inequality follows from Fact 3.5 applied to the sets
U =
X
S1 ∈
2
: |NΓ (S1 , Y )| > 2p|Y |
and Y with ξ = 1. The second inequality follows from thechoice of β , the third one follows from |Y | > σ n, and the last one holds because |X | > σ n and γ 6 σ 3 δ ′ /12. We want to bound |B2 | from above. By definition, if a pair of vertices belongs to B2 , then it X does not belong to B1 . Then, consider a pair of vertices S1 ∈ 2 such that |NΓ (S1 , Y )| 6 2p|Y |. ′ Consider a set Y ⊂ Y of size exactly 2p|Y | that contains NΓ (S1 , Y ). Applying Fact 3.5 to the sets
: |NΓ (S2 , Y ′ )| > 2p|Y ′ | and Y ′ with ξ = 1, we conclude that there are at most β 2 /(p2 |Y ′ |) X pairs S2 ∈ 2 such that |NΓ (S1 , S2 , Y )| > 4p2 |Y |. Therefore,
S2 ∈
X 2
2 γ 2 n2 p δ ′ |X | 6 p . |B2 | 6 2 2σ 6 2 2 X X The summation below is over the pairs (S1 , S2 ) ∈ 2 × 2 \ (B1 ∪ B2 ). By (16) and the upper
|X |
β2 6 p2 2p|Y |
|X |
bounds on B1 and B2 we conclude that
2 |NG (S1 , S2 ; Y )| − q2 |Y | > δ ′ p2 |X | |Y | − |B1 | |Y | − |B2 |2p|Y | 2
> δ ′ p2
=
δ′ 3
2
p
|X |
2
2
|X | 2
|Y | −
2δ ′ 3
p2
|X |
2
2
|Y |
2 |Y |.
(17)
The contribution of the pairs (S1 , S2 ) ̸∈ (B1 ∪ B2 ) with |NG (S1 , S2 ; Y )| − q2 |Y | 6 δ ′ q2 |Y |/6 to the sum in (17) is at most
δ′ 6
p2
|X | 2
2 |Y |.
Note that, by the definition of B2 , for all (S1 , S2 ) ̸∈ (B1 ∪ B2 ), the following holds.
|NG (S1 , S2 ; Y )| − q2 |Y | 6 max q2 |Y |, (4p2 − q2 )|Y | 6 4p2 |Y |.
(18)
12
Y. Kohayakawa et al. / European Journal of Combinatorics (
Hence, by (17) and (18), there exist at least δ ′
|X |
2
2
)
/24 pairs (S1 , S2 ) ∈
–
X 2
×
X 2
\ (B1 ∪ B2 ) such
that ′ |NG (S1 , S2 ; Y )| − q2 |Y | > δ q2 |Y | = ξ q2 |Y |.
(19)
6
Now let us define two auxiliary graphs B+ and B− with vertex-set
E (B+ ) =
E (B− ) =
X 2
{S1 , S2 } ∈
:
2
X 2
{S1 , S2 } ∈
Since there are at least δ ′
|X |
2
2
:
2
|X |
2
max{e(B+ ), e(B− )} >
4
X 2
and edge-sets as follows.
(1 + ξ )q2 |Y | < |NG (S1 , S2 ; Y )| 6 4p2 |Y | |NG (S1 , S2 ; Y )| < (1 − ξ )q2 |Y | .
/24 pairs (S1 , S2 ) ∈
2
δ ′ /24
−
|X |
2
X 2
×
X 2
> ϱ0
|X | 2
2
such that (19) holds, we have
,
where in the first inequality the term ‘‘4’’ in the denominator comes from the fact that now we are counting unordered pairs and the edges belong either to E (B+ ) or E (B− ). Furthermore, we discount the pairs {S1 , S1 }. Suppose without lost of generality that e(B+ ) > ϱ0
|X |
2
. Then, Fact 3.9 implies that there exist
2
subsets X1 , X2 ⊂ 2 with |X1 |, |X2 | > ζ 2 such that (X1 , X2 )B+ satisfies DISC(ϱ, 1, µ) for some ϱ > ϱ0 . Recall that Γ = (V , EΓ) isa 3-uniform (p, β)-jumbled hypergraph with n vertices. By Fact 3.10, X
the bipartite graph Γ ′ = ( EΓ ′
V 2
|X |
, V ; EΓ ′ ), where
= {v1 , v2 }, v : {v1 , v2 } ∈ V2 , v ∈ V and {v1 , v2 , v} ∈ EΓ
is a (p, β)-jumbled graph. Note that X1 , X2 ⊂
X 2
⊂
V 2
with |X1 |, |X2 | > ζ
|X |
2
> ζ
σn 2
> (ζ σ 2 /2) 2 > σ ′ 2 . Therefore, the hypotheses of Lemma 3.8 are satisfied. By Lemma 3.8 we conclude that for all but at most ν|Y | vertices y ∈ Y , the pair (NΓ ′ (y, X1 ), NΓ ′ (y, X2 ))B+ satisfies DISC(ϱ, 1, ν), which implies the following statement for all but at most ν|Y | vertices y ∈ Y .
n
n
(NΓ (y, X1 ), NΓ (y, X2 ))B+ satisfies DISC(ϱ, 1, ν).
(20)
Now let us estimate the number of triplets (S1 , S2 , y) in X1 × X2 × Y such that {S1 , S2 } is an edge of B+ and the pairs S1 and S2 belong to the neighbourhood of y in G. Formally, we define such triplets as follows.
T = {(S1 , S2 , y) ∈ X1 × X2 × Y : S1 ∈ NG (y, X1 ), S2 ∈ NG (y, X2 ), {S1 , S2 } ∈ E (B+ )}. By the definition of B+ we have
|T | > (1 + ξ )q2 |Y |eB+ (X1 , X2 ) > (1 + ξ )q2 |Y |(ϱ − µ)|X1 | |X2 | ξ > 1+ ϱq2 |X1 | |X2 | |Y |, 2
(21)
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where in the second inequality we used the fact that (X1 , X2 )B+ ∈ DISC(ϱ, 1, µ) and the last one follows from the choice of µ. + NG (y, X1 ), NG (y, X2 ) . Now we give an upper bound on |T | contradicting (21). We have |T | = y∈Y eB Put Y ′ = y ∈ Y : dΓ (y, Xi ) 6 2p|Xi | for both i = 1, 2 .
By (20), for all but at most ν|Y | vertices y ∈ Y ′ we have eB+ (NG (y, X1 ), NG (y, X2 )) 6 ϱNG (y, X1 ) |NG (y, X2 )| + ν|NΓ (y, X1 )| |NΓ (y, X2 )|
6 ϱ dG (y, X1 )dG (y, X2 ) + 4ν p2 |X1 | |X2 |.
The last inequality follows from the fact that y ∈ Y ′ . Now we willbound the terms related to vertices in Y \ Y ′ . By Fact 3.5, we have |Y \ Y ′ | 6 β 2 1/p2 |X1 | + 1/p2 |X2 | . Then,
ϱ dG (y, X1 )dG (y, X2 ) + 4ν p2 |X1 | |X2 | + ν|Y |4p2 |X1 | |X2 |
|T | 6
y∈Y ′
+
β2 β2 + |X1 | |X2 |. p2 |X1 | p2 |X2 |
The next inequality is obtained by putting the following facts together: ν = α 2 ξ ϱ/64, β 6 γ p2 n3/2 , √ ′ n ′ γ 6 (α/2) ξ ϱ0 σ σ /24, q > α p, |X1 |, |X2 | > σ 2 and |Y | > σ n.
|T | 6 ϱ
(dG (y, X1 )dG (y, X2 )) +
y∈Y ′
ξ 4
ϱq2 |X1 | |X2 | |Y |.
(22)
√ ε ′ )q|Xi | for both i = 1, 2. Since (X , Y )G ∈ DISC(q, p, ε′ ), it is √ |X | |X | not hard to see that |Yi′′ | 6 ε ′ p 2 |Y |/(q|Xi |) for both i = 1, 2. Since |X1 |, |X2 | > ζ 2 , q > α p Define Yi′′ = y ∈ Y : dG (y, Xi ) > (1 +
and ε ′ 6 (α 3 ξ ϱ0 ζ /64)2 , the following holds for both i = 1, 2.
|Yi′′ | 6
ξ ϱα 2 64
|Y |.
Note that
(dG (y, X1 )dG (y, X2 )) =
(dG (y, X1 )dG (y, X2 ))
y∈Y ′ \(Y1′′ ∪Y2′′ )
y∈Y ′
+
(dG (y, X1 )dG (y, X2 ))
y∈Y ′ ∩(Y1′′ ∪Y2′′ )
6 |Y |(1 +
√ 2 2 ξ ϱα 2 |Y | 4p2 |X1 | |X2 | . ε ′ ) q |X1 | |X2 | + 32
√ Therefore, since ε ′ 6 ξ /24, the above inequality together with (22) implies ξ |T | 6 1 + ϱq2 |X1 | |X2 | |Y |, 2
a contradiction with (21).
4. Proof of the main result In this section we show how to combine the lemmas presented in Section 3 in order to prove Theorem 1.2.
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Proof of Theorem 1.2. Let ε , α , η > 0, C > 1 and k > 4 be given. Let H1 , . . . , Hr be all the k-vertex 3-uniform hypergraphs which are linear and connector-free. Applying Lemma 3.1 with parameters k, C , ε and d = k for H1 , . . . , Hr , we obtain, respectively, constants δ1 , . . . , δr . Now put δmin = min{δ1 , . . . , δr }. Let δ ′ be given by Lemma 2.4 applied with α and δmin . Let ε ′ be given by Lemma 3.4 applied with α and δ ′ . Lemma 3.3 applied with ε ′ , η and σ = 1 gives δ . Following the quantification of Lemma 3.3 applied with α we obtain γ1 . Finally, following the quantification of Lemma 3.4 applied with σ = 1 we obtain γ2 . Put γ = min{γ1 , γ2 }. Let p = p(n) = o(1) with p ≫ n−2/(k−1) and let q = q(n) be such that α p 6 q 6 p. In what follows we suppose that n is sufficiently large. Let Γ = (V , EΓ ) be an n-vertex (p, β)-jumbled 3-uniform hypergraph and let G be a spanning n subhypergraph of Γ with |E (G)| = q 3 such that G satisfies Q′ (η, δ, q) and BDD(k, C , q). Suppose that β 6 γ p2 n3/2 . We want to prove that G contains (1 ± ε)nk q|E (H )| copies of all linear 3-uniform connector-free hypergraphs H with k vertices. By Lemma 3.3, our hypergraph G satisfies DISC(q, p, ε ′ ). Now apply Lemmas 2.4 and 3.4 in succession to deduce that G satisfies PAIR(q, p, δ ′ ) and TUPLE(δ, q). Now let H be any linear 3-uniform connector-free hypergraphs H with k vertices. Since G satisfies TUPLE(δ, q) and BDD(k, C , q), by Lemma 3.1, we conclude that
|Emb(H , G)| − nk q|E (H )| < εnk q|E (H )| . Acknowledgements We thank the referees for their helpful suggestions and comments. Y. Kohayakawa and G. O. Mota were partially supported by FAPESP (2013/03447-6) and CNPq (459335/2014-6). Y. Kohayakawa was also supported by FAPESP (2013/07699-0), CNPq (310974/2013-5), NUMEC/USP (Project MaCLinC/USP) and the NSF (DMS 1102086). G. O. Mota was also supported by FAPESP (2013/114312, 2013/20733-2). M. Schacht was supported by the Heisenberg-Programme of the DFG (grant SCHA 1263/4-1). A. Taraz was supported in part by DFG grant TA 309/2-2. This research was supported by a joint CAPES/DAAD PROBRAL project (Proj. 430/15). References [1] B. Bollobás, Extremal Graph Theory, in: London Mathematical Society Monographs, vol. 11, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978, MR506522. [2] F. Chung, R. Graham, Sparse quasi-random graphs, Combinatorica 22 (2) (2002) 217–244. http://dx.doi.org/10.1007/ s004930200010, Special issue: Paul Erdős and his mathematics, MR1909084 (2003d:05110). [3] F. Chung, R. Graham, Quasi-random graphs with given degree sequences, Random Struct. Algorithms 32 (1) (2008) 1–19. http://dx.doi.org/10.1002/rsa.20188, MR2371048 (2009a:05189). [4] F. Chung, R. Graham, R. Wilson, Quasi-random graphs, Combinatorica 9 (4) (1989) 345–362. http://dx.doi.org/10.1007/ BF02125347, MR1054011. [5] P. Erdős, Some old and new problems in various branches of combinatorics, in: Proc. 10th Southeastern Conference on Combinatorics, Graph Theory and Computing, 1979, pp. 19–37. [6] Y. Kohayakawa, G.O. Mota, M. Schacht, A. Taraz, Counting results for sparse pseudorandom hypergraphs I, European J. Combin. (2017) http://dx.doi.org/10.1016/j.ejc.2017.04.008, (in press). [7] Y. Kohayakawa, V. Rödl, Szemerédi’s regularity lemma and quasi-randomness, in: Recent Advances in Algorithms and Combinatorics, 2003, pp. 289–351. http://dx.doi.org/10.1007/0-387-22444-0_9, MR1952989 (2003j:05065). [8] Y. Kohayakawa, V. Rödl, M. Schacht, J. Skokan, On the triangle removal lemma for subgraphs of sparse pseudorandom graphs, in: An Irregular Mind, 2010, pp. 359–404. http://dx.doi.org/10.1007/978-3-642-14444-8_10, MR2815608. [9] M. Krivelevich, B. Sudakov, Pseudo-random graphs, in: More Sets, Graphs and Numbers, 2006, pp. 199–262. http://dx.doi. org/10.1007/978-3-540-32439-3_10, MR2223394 (2007a:05130). [10] J. Lenz, D. Mubayi, The poset of hypergraph quasirandomness, Random Struct. Algorithms 46 (4) (2015) 762–800. http:// dx.doi.org/10.1002/rsa.20521, MR3346466. [11] V. Nikiforov, On the edge distribution of a graph, Combin. Probab. Comput. 10 (6) (2001) 543–555. http://dx.doi.org/10. 1017/S0963548301004837, MR1869845. [12] Chr. Reiher, V. Rödl, M. Schacht, Embedding tetrahedra into quasirandom hypergraphs, ArXiv e-prints, Feb. 2016, Availble at arXiv:1602.02289. [13] V. Rödl, On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1–2) (1986) 125–134. MR837962 (88b:05098). [14] A. Thomason, Pseudorandom graphs, in: Random Graphs’85, (Poznań, 1985), 1987, pp. 307–331. MR89d:05158. [15] A. Thomason, Random graphs, strongly regular graphs and pseudorandom graphs, in: Surveys in Combinatorics 1987, (New Cross, 1987), 1987, pp. 173–195. MR88m:05072. [16] H. Towsner, σ -algebras for quasirandom hypergraphs, Random Struct. Algorithms 50 (1) (2017) 114–139. http://dx.doi. org/10.1002/rsa.20641.