- Email: [email protected]
An extension of the Erdős–Ko–Rado Theorem
Electronic Notes in Discrete Mathematics 28 (2007) 117–123 www.elsevier.com/locate/endm
An extension of the Erd˝os–Ko–Rado Theorem Norihide Tokushige 1,2 College of Education Ryuku University Okinawa, JAPAN
Abstract A family G ⊂ 2[n] is called r-wise t-intersecting if |G1 ∩ · · · ∩ Gr | ≥ t holds for all G1 , . . . , Gr ∈ G. We report some results concerning the maximum size of r-wise tintersecting families with additional conditions. For any k-uniform r-wise example, t-intersecting family on 2k vertices has size at most 2k−1 for r ≥ 7, 1 ≤ t ≤ 2r −r−1 k−1 and k > k0 (r). Keywords: Erd˝ os–Ko–Rado theorem, intersecting families.
1
Introduction
A family (or hypergraph) G ⊂ 2[n] is called r-wise t-intersecting if |G1 ∩ · · · ∩ Gr | ≥ t holds for all G1 , . . . , Gr ∈ G. The aim of this note is to report some results about the largest r-wise t-intersecting families with additional conditions. Let us define a typical r-wise t-intersecting family Gi (n, r, t) and its k-uniform subfamily Fi (n, k, r, t) as follows: Gi (n, r, t) = {G ⊂ [n] : |G ∩ [t + ri]| ≥ t + (r − 1)i}, Fi (n, k, r, t) = Gi (n, r, t) ∩ [n] . k 1 2
The author was supported by MEXT Grant-in-Aid for Scientific Research (B) 16340027. Email: [email protected]
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.017
118
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
An r-wise t-intersecting family G is called non-trivial if | G| < t, where [n] G := G∈G G. Two families G, G ⊂ 2 are said to be isomorphic and denoted by G ∼ = G if there exists a vertex permutation τ on [n] such that G = {{τ (g) : g ∈ G} : G ∈ G}. Let m(n, k, r, t) be the maximal size of k-uniform r-wise t-intersecting families on n vertices. To determine m(n, k, r, t) is one of the oldest problems in extremal set theory, which is still widely open. The case r = 2 was observed by Erd˝os–Ko–Rado[6], Frankl[9], Wilson[31], and then m(n, k, 2, t) = maxi |Fi (n, k, 2, t)| was finally proved by Ahlswede and Khachatrian[2]. On the other hand, only a few results are known for the case r ≥ 3. Frankl[8] showed m(n, k, r, 1) = |F0 (n, k, r, 1)| if (r − 1)n ≥ rk. Partial results for the cases r ≥ 3 and t ≥ 2 are found in [12,14,26,27,28,30]. All known results suggest m(n, k, r, t) = max |Fi (n, k, r, t)| i
in general, and we will consider the case when the maximum is attained by F0 or F1 . To state our result let us define a list A of acceptable parameters as follows. (1)
A = {(r, t) : r ≥ 5, 1 ≤ t ≤ 2r+1 − 3r − 1} − {(5, 2), (5, 3), (5, 4), (6, 2)}.
Theorem 1.1 Let (r, t) ∈ A be fixed. Then there exist positive constants , n0 such that m(n, k, r, t) = max{|F0 (n, k, r, t)|, |F1 (n, k, r, t)|} holds for all n > n0 and k with | nk − 12 | < . n−t−r We note that |F0 (n, k, r, t)| = n−t and |F1 (n, k, r, t)| = (t + r) k−t−r+1 + k−t n−t−r . If (r, t) ∈ A and r k then max{|F0 |, |F1 |} is attained by k−t−r ⎧ ⎪ ⎪ F (n, k, r, t) if 1 ≤ t ≤ 2r − r − 2, ⎪ ⎨ 0
or t = 2r − r − 1 and n ≥ 2k − 2r + r/2 + 3, ⎪ ⎪ ⎪ ⎩ F (n, k, r, t) if t ≥ 2r − r, or t = 2r − r − 1 and n ≤ 2k − 2r + r/2 + 2. 1 Conjecture 1.2 Theorem 1.1 is true for all r ≥ 3 and 1 ≤ t ≤ 2r+1 − 3r − 1.
2
Non-trivial intersecting families
Let m∗ (n, k, r, t) be the maximal size of non-trivial n-vertex k-uniform r-wise t-intersecting families. Ahlswede and Khachatrian[1] determined m∗ (n, k, 2, t) completely, which included earlier results of Hilton–Milner[21] and Frankl[10].
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
119
In [29] a k-uniform version of the Brace–Daykin theorem[4] is considered for m∗ (n, k, r ≥ 8, 1) and k/n ≈ 1/2. To state our result let us define some families of k-uniform hypergraphs as follows. F(n, k, r, t) = {F ⊂ [n] : F is r-wise t-intersecting}, k [n] Fj (n, k, r, t) = {F ⊂ k : F ⊂ F for some F ∼ = Fj (n, k, r, t)}, Yi (n, k, r, t) = F(n, k, r, t) − Fj (n, k, r, t). 0≤j≤i
For fixed n, k, r, t, we clearly have Fj ⊂ F. We are interested in m∗ = max{|F| : F ∈ Y0 }. It seems that hypergraphs in F with nearly largest size only come from some Fj , moreover they are stable in a sense, namely, max{|F| : F ∈ Y1 } < (1 − γ)m∗ for some fixed constant γ > 0. We verify this phenomenon in the case t ≤ 2r+1 − 3r − 1 and k/n ≈ 1/2. See [16,24] for more about stability type results. Define a list A∗ of acceptable parameters as follows: (2)
A∗ = A ∪ {(5, 1), (6, 1), (7, 1)},
where A is defined by (1). The following is our main result, which immediately implies Theorem 1.1. Theorem 2.1 Let (r, t) ∈ A∗ be fixed. Then there exist positive constants γ, , n0 such that the following (i) and (ii) are true for all n > n0 and k with | nk − 12 | < . (i) m∗ (n, k, r, t) = |F1 (n, k, r, t)|. (ii) If F ∈ Y1 (n, k, r, t) then |F| < (1 − γ) m∗ (n, k, r, t).
3
Intersecting Sperner families
We apply Theorem 2.1 to get a Sperner type inequality[25]. A family G ⊂ 2[n] is called a Sperner family if G ⊂ G holds for all distinct G, G ∈ G. Let s(n, r, t) be the maximal size of r-wise t-intersecting n Sperner families on n vertices. Milner[23] proved s(n, r = 2, t) = (n+t)/2 . Frankl[8] and Gronau[17,18,19,20] determined s(n,r = 3, t = 1) for n ≥ 53. Gronau[18] also n−1 proved s(n, r ≥ 4, t = 1) = (n−1)/2 for all n. For sufficiently large n, it was n−2 n−t proved that s(n, r ≥ 4, t = 2) = (n−2)/2 for in [12], s(n, r, t) = (n−t)/2 r−2 r ≥ 5 and 1 ≤ t ≤ 2 log 2 − 1 in [30], and s(n, r = 3, t = 2) was determined in [14]. Using Theorem 2.1 we get the following.
120
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
Theorem 3.1 Let r ≥ 6 and 1 ≤ t ≤ 2r+1 − 3r − 1. Then there exists n0 such that ⎧ ⎨ |F (n, k , r, t)| if 1 ≤ t ≤ 2r − r − 2 0 0 s(n, r, t) = ⎩ |F (n, k , r, t)| if 2r − r − 1 ≤ t ≤ 2r+1 − 3r − 1 1 1
, t + n−t } and k1 = t + r − 1 + n−t−r
. for all n > n0 , where k0 ∈ {t + n−t 2 2 2 Moreover F0 (n, k0 , r, t) and F1 (n, k1 , r, t) are the only optimal configurations (up to isomorphism). Conjecture 3.2 Theorem 3.1 is true for r = 4, 5 as well. Due to the results mentioned above [18,12], the conjecture is true for t = 1, 2. Our proof of Theorem 3.1 is valid for all (r, t) ∈ A, and the conjecture is open for “r = 4 and 3 ≤ t ≤ 19” and “r = 5 and t = 3, 4.” The conjecture n−1 fails for r = 3. In fact it is known from [8,17,14] that s(n = 2m, 3, 1) = + 1, m n−2 The exact value of s(n = 2m + 1, 3, 2) = m + 2 (for n large enough). n−3 s(n, 3, 3) is not known, while s(n = 2m, 3, 3) ≥ m−1 + 3.
4
The p-weight version
We introduce a weighted version of Frankl’s result in [11]. Let p and q = 1 − p denote positive real numbers. For a family G ⊂ 2[n] we define the p-weight of G, denoted by wp (G), as follows: wp (G) =
p|G| q n−|G| =
n
G ∩ X pi q n−i . i=0
G∈G
i
Let w(n, p, r, t) be the maximal p-weight of r-wise t-intersecting families on n vertices, and let w∗ (n, p, r, t) be the maximal p-weight of non-trivial r-wise t-intersecting families on n vertices. It might be natural to expect w(n, p, r, t) = max wp (Gi (n, r, t)). i
Ahlswede and Khachatrian proved that this is true for r = 2 in [3] (cf. [7,30,5]). This includes the Katona theorem[22] about w(n, 1/2, 2, t). It is shown in [13] that w(n, p, r, 1) = wp (G0 (n, r, 1)) = p for p ≤ (r − 1)/r.
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
121
Partial results for w∗ (n, p, r, 1) are found in [15,29], which extend the result of Brace–Daykin[4]: w∗ (n, 1/2, r, 1) = w1/2 (G1 (n, r, 1)). To state our result let us define some families of hypergraphs as follows. G(n, r, t) = {G ⊂ 2[n] : G is r-wise t-intersecting}, Gj (n, r, t) = {G ⊂ 2[n] : G ⊂ G for some G ∼ = Gj (n, r, t)}, Gj (n, r, t). Xi (n, r, t) = G(n, r, t) − 0≤j≤i
Theorem 4.1 Let (r, t) ∈ A∗ be fixed, where A∗ is defined by (2). Then there exist positive constants γ, such that the following (i) and (ii) are true for all n ≥ r + t and p with |p − 12 | < . (i) w∗ (n, p, r, t) = wp (G1 (n, r, t)). (ii) If G ∈ X1 (n, r, t) then wp (G) < (1 − γ) w∗ (n, p, r, t). ∗ In [15] it is shown √ by construction that w (n, p, 5, 1) > wp (G1 (n, 5, 1)) for all 1/2 < p < (1 + 21)/10. Theorem 4.1 could be true for all r ≥ 5 with only exception r = 5 and t = 1, and the same extension could be expected for Theorem 2.1. The upper bound for t set by (2) in Theorem 4.1 is best possible. In fact we have wp (G2 (n, r, t)) > wp (G1 (n, r, t)) for t > 2r+1 − 3r − 1. We emphasize that Frankl has already got a special case of (i) of Theorem 4.1 in [11] (Theorem 6.4), where he proved
w∗ (n, 1/2, r, t) = w1/2 (G1 (n, r, t)) for r ≥ 5 and 1 ≤ t ≤ 2r − r − 1. Theorem 4.1 implies the following immediately. Theorem 4.2 Let (r, t) ∈ A be fixed, where A is defined by (1). Then there exists positive constant such that w(n, p, r, t) = max{wp (G0 (n, r, t)), wp (G1 (n, r, t))} holds for all n ≥ r + t and p with |p − 12 | < . We find that if (r, t) ∈ A then max{wp (G1 ), wp (G2 )} is attained by ⎧ ⎨ G (n, r, t) if 1 ≤ t ≤ 2r − r − 2, or t = 2r − r − 1 and p ≤ 1/2, 0 ⎩ G (n, r, t) if t ≥ 2r − r, or t = 2r − r − 1 and p > 1/2. 1
122
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
References [1] R. Ahlswede, L.H. Khachatrian. The complete nontrivial-intersection theorem for systems of finite sets. J. Combin. Theory (A), 76:121-138, 1996. [2] R. Ahlswede, L.H. Khachatrian. The complete intersection theorem for systems of finite sets. European J. Combin., 18:125–136, 1997. [3] R. Ahlswede, L.H. Khachatrian. The diametric theorem in Hamming spaces — Optimal anticodes. Adv. in Appl. Math., 20:429–449, 1998. [4] A. Brace, D. E. Daykin. A finite set covering theorem. Bull. Austral. Math. Soc., 5:197–202, 1971. [5] C. Bey, K. Engel. Old and new results for the weighted t-intersection problem via AK-methods. Numbers, Information and Complexity, Althofer, Ingo, Eds. et al., Dordrecht, Kluwer Academic Publishers, 45–74, 2000. [6] P. Erd˝ os, C. Ko, R. Rado. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford (2), 12:313–320, 1961. [7] I. Dinur, S. Safra. On the Hardness of Approximating Minimum Vertex-Cover. Annals of Mathematics, 162:439-485, 2005. [8] P. Frankl. On Sperner families satisfying an additional condition. J. Combin. Theory (A), 20:1–11, 1976. [9] P. Frankl. The Erd˝ os–Ko–Rado theorem is true for n = ckt. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthey, 1976), Vol. I, 365–375, Colloq. math. Soc. J´ anos Bolyai, 18, North–Holland, 1978. [10] P. Frankl. On intersecting families of finite sets. J. Combin. Theory (A), 24:141–161, 1978. [11] P. Frankl. Multiply-intersecting families. J. Combin. Theory (B), 53:195–234, 1991. [12] P. Frankl, N. Tokushige. Weighted 3-wise 2-intersecting families. J. Combin. Theory (A) 100:94–115, 2002. [13] P. Frankl, N. Tokushige. Weighted multiply intersecting families. Studia Sci. Math. Hungarica 40:287–291, 2003. [14] P. Frankl, N. Tokushige. Random walks and multiply intersecting families. J. Combin. Theory (A), 109:121-134, 2005. [15] P. Frankl, N. Tokushige. Weighted non-trivial multiply intersecting families. Combinatorica, 26 37–46, 2006.
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
123
[16] E. Friedgut. On the measure of intersecting families, uniqueness and stability. preprint. [17] H.-D.O.F. Gronau. On Sperner families in which no 3-sets have an empty intersection. Acta Cybernet., 4:213–220, 1978. [18] H.-D.O.F. Gronau. On Sperner families in which no k-sets have an empty intersection. J. Combin. Theory (A), 28:54–63, 1980. [19] H.-D.O.F. Gronau. On Sperner families in which no k-sets have an empty intersection II. J. Combin. Theory (A), 30:298–316, 1981. [20] H.-D.O.F. Gronau. On Sperner families in which no k-sets have an empty intersection III. Combinatorica, 2:25–36, 1982. [21] A. J. W. Hilton, E. C. Milner. Some intersection theorems for systems of finite sets. Quart. J. Math. Oxford, 18:369–384, 1967. [22] G.O.H. Katona. Intersection theorems for systems of finite sets. Acta Math. Acad. Sci. Hung., 15:329–337, 1964. [23] E.C. Milner. A combinatorial theorem on systems of sets. J. London Math. Soc., 43:204–206, 1968. [24] D. Mubayi. A stability method in extremal set theory. preprint. [25] E. Sperner. Ein Satz u ¨ber Untermengen einer endlichen Mente. Zeitschrift, 27:544–548, 1928.
Math.
[26] N. Tokushige. Extending the Erd˝ os–Ko–Rado theorem. J. Combin. Designs, 14:52–55, 2006. [27] N. Tokushige. The maximum size of 3-wise t-intersecting families. European J. Combin, in press. [28] N. Tokushige. EKR type inequalities for 4-wise intersecting families. to appear in J. Combin. Theory (A). [29] N. Tokushige. preprint.
Brace–Daykin type inequalities for intersecting families.
[30] N. Tokushige. Intersecting families — uniform versus weighted. Ryukyu Math. J., 18:89–103, 2005. [31] R.M. Wilson. The exact bound in the Erd˝ os–Ko–Rado theorem. Combinatorica, 4:247–257, 1984.
An extension of the Erd˝os–Ko–Rado Theorem Norihide Tokushige 1,2 College of Education Ryuku University Okinawa, JAPAN
Abstract A family G ⊂ 2[n] is called r-wise t-intersecting if |G1 ∩ · · · ∩ Gr | ≥ t holds for all G1 , . . . , Gr ∈ G. We report some results concerning the maximum size of r-wise tintersecting families with additional conditions. For any k-uniform r-wise example, t-intersecting family on 2k vertices has size at most 2k−1 for r ≥ 7, 1 ≤ t ≤ 2r −r−1 k−1 and k > k0 (r). Keywords: Erd˝ os–Ko–Rado theorem, intersecting families.
1
Introduction
A family (or hypergraph) G ⊂ 2[n] is called r-wise t-intersecting if |G1 ∩ · · · ∩ Gr | ≥ t holds for all G1 , . . . , Gr ∈ G. The aim of this note is to report some results about the largest r-wise t-intersecting families with additional conditions. Let us define a typical r-wise t-intersecting family Gi (n, r, t) and its k-uniform subfamily Fi (n, k, r, t) as follows: Gi (n, r, t) = {G ⊂ [n] : |G ∩ [t + ri]| ≥ t + (r − 1)i}, Fi (n, k, r, t) = Gi (n, r, t) ∩ [n] . k 1 2
The author was supported by MEXT Grant-in-Aid for Scientific Research (B) 16340027. Email: [email protected]
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.017
118
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
An r-wise t-intersecting family G is called non-trivial if | G| < t, where [n] G := G∈G G. Two families G, G ⊂ 2 are said to be isomorphic and denoted by G ∼ = G if there exists a vertex permutation τ on [n] such that G = {{τ (g) : g ∈ G} : G ∈ G}. Let m(n, k, r, t) be the maximal size of k-uniform r-wise t-intersecting families on n vertices. To determine m(n, k, r, t) is one of the oldest problems in extremal set theory, which is still widely open. The case r = 2 was observed by Erd˝os–Ko–Rado[6], Frankl[9], Wilson[31], and then m(n, k, 2, t) = maxi |Fi (n, k, 2, t)| was finally proved by Ahlswede and Khachatrian[2]. On the other hand, only a few results are known for the case r ≥ 3. Frankl[8] showed m(n, k, r, 1) = |F0 (n, k, r, 1)| if (r − 1)n ≥ rk. Partial results for the cases r ≥ 3 and t ≥ 2 are found in [12,14,26,27,28,30]. All known results suggest m(n, k, r, t) = max |Fi (n, k, r, t)| i
in general, and we will consider the case when the maximum is attained by F0 or F1 . To state our result let us define a list A of acceptable parameters as follows. (1)
A = {(r, t) : r ≥ 5, 1 ≤ t ≤ 2r+1 − 3r − 1} − {(5, 2), (5, 3), (5, 4), (6, 2)}.
Theorem 1.1 Let (r, t) ∈ A be fixed. Then there exist positive constants , n0 such that m(n, k, r, t) = max{|F0 (n, k, r, t)|, |F1 (n, k, r, t)|} holds for all n > n0 and k with | nk − 12 | < . n−t−r We note that |F0 (n, k, r, t)| = n−t and |F1 (n, k, r, t)| = (t + r) k−t−r+1 + k−t n−t−r . If (r, t) ∈ A and r k then max{|F0 |, |F1 |} is attained by k−t−r ⎧ ⎪ ⎪ F (n, k, r, t) if 1 ≤ t ≤ 2r − r − 2, ⎪ ⎨ 0
or t = 2r − r − 1 and n ≥ 2k − 2r + r/2 + 3, ⎪ ⎪ ⎪ ⎩ F (n, k, r, t) if t ≥ 2r − r, or t = 2r − r − 1 and n ≤ 2k − 2r + r/2 + 2. 1 Conjecture 1.2 Theorem 1.1 is true for all r ≥ 3 and 1 ≤ t ≤ 2r+1 − 3r − 1.
2
Non-trivial intersecting families
Let m∗ (n, k, r, t) be the maximal size of non-trivial n-vertex k-uniform r-wise t-intersecting families. Ahlswede and Khachatrian[1] determined m∗ (n, k, 2, t) completely, which included earlier results of Hilton–Milner[21] and Frankl[10].
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
119
In [29] a k-uniform version of the Brace–Daykin theorem[4] is considered for m∗ (n, k, r ≥ 8, 1) and k/n ≈ 1/2. To state our result let us define some families of k-uniform hypergraphs as follows. F(n, k, r, t) = {F ⊂ [n] : F is r-wise t-intersecting}, k [n] Fj (n, k, r, t) = {F ⊂ k : F ⊂ F for some F ∼ = Fj (n, k, r, t)}, Yi (n, k, r, t) = F(n, k, r, t) − Fj (n, k, r, t). 0≤j≤i
For fixed n, k, r, t, we clearly have Fj ⊂ F. We are interested in m∗ = max{|F| : F ∈ Y0 }. It seems that hypergraphs in F with nearly largest size only come from some Fj , moreover they are stable in a sense, namely, max{|F| : F ∈ Y1 } < (1 − γ)m∗ for some fixed constant γ > 0. We verify this phenomenon in the case t ≤ 2r+1 − 3r − 1 and k/n ≈ 1/2. See [16,24] for more about stability type results. Define a list A∗ of acceptable parameters as follows: (2)
A∗ = A ∪ {(5, 1), (6, 1), (7, 1)},
where A is defined by (1). The following is our main result, which immediately implies Theorem 1.1. Theorem 2.1 Let (r, t) ∈ A∗ be fixed. Then there exist positive constants γ, , n0 such that the following (i) and (ii) are true for all n > n0 and k with | nk − 12 | < . (i) m∗ (n, k, r, t) = |F1 (n, k, r, t)|. (ii) If F ∈ Y1 (n, k, r, t) then |F| < (1 − γ) m∗ (n, k, r, t).
3
Intersecting Sperner families
We apply Theorem 2.1 to get a Sperner type inequality[25]. A family G ⊂ 2[n] is called a Sperner family if G ⊂ G holds for all distinct G, G ∈ G. Let s(n, r, t) be the maximal size of r-wise t-intersecting n Sperner families on n vertices. Milner[23] proved s(n, r = 2, t) = (n+t)/2 . Frankl[8] and Gronau[17,18,19,20] determined s(n,r = 3, t = 1) for n ≥ 53. Gronau[18] also n−1 proved s(n, r ≥ 4, t = 1) = (n−1)/2 for all n. For sufficiently large n, it was n−2 n−t proved that s(n, r ≥ 4, t = 2) = (n−2)/2 for in [12], s(n, r, t) = (n−t)/2 r−2 r ≥ 5 and 1 ≤ t ≤ 2 log 2 − 1 in [30], and s(n, r = 3, t = 2) was determined in [14]. Using Theorem 2.1 we get the following.
120
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
Theorem 3.1 Let r ≥ 6 and 1 ≤ t ≤ 2r+1 − 3r − 1. Then there exists n0 such that ⎧ ⎨ |F (n, k , r, t)| if 1 ≤ t ≤ 2r − r − 2 0 0 s(n, r, t) = ⎩ |F (n, k , r, t)| if 2r − r − 1 ≤ t ≤ 2r+1 − 3r − 1 1 1
, t + n−t } and k1 = t + r − 1 + n−t−r
. for all n > n0 , where k0 ∈ {t + n−t 2 2 2 Moreover F0 (n, k0 , r, t) and F1 (n, k1 , r, t) are the only optimal configurations (up to isomorphism). Conjecture 3.2 Theorem 3.1 is true for r = 4, 5 as well. Due to the results mentioned above [18,12], the conjecture is true for t = 1, 2. Our proof of Theorem 3.1 is valid for all (r, t) ∈ A, and the conjecture is open for “r = 4 and 3 ≤ t ≤ 19” and “r = 5 and t = 3, 4.” The conjecture n−1 fails for r = 3. In fact it is known from [8,17,14] that s(n = 2m, 3, 1) = + 1, m n−2 The exact value of s(n = 2m + 1, 3, 2) = m + 2 (for n large enough). n−3 s(n, 3, 3) is not known, while s(n = 2m, 3, 3) ≥ m−1 + 3.
4
The p-weight version
We introduce a weighted version of Frankl’s result in [11]. Let p and q = 1 − p denote positive real numbers. For a family G ⊂ 2[n] we define the p-weight of G, denoted by wp (G), as follows: wp (G) =
p|G| q n−|G| =
n
G ∩ X pi q n−i . i=0
G∈G
i
Let w(n, p, r, t) be the maximal p-weight of r-wise t-intersecting families on n vertices, and let w∗ (n, p, r, t) be the maximal p-weight of non-trivial r-wise t-intersecting families on n vertices. It might be natural to expect w(n, p, r, t) = max wp (Gi (n, r, t)). i
Ahlswede and Khachatrian proved that this is true for r = 2 in [3] (cf. [7,30,5]). This includes the Katona theorem[22] about w(n, 1/2, 2, t). It is shown in [13] that w(n, p, r, 1) = wp (G0 (n, r, 1)) = p for p ≤ (r − 1)/r.
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
121
Partial results for w∗ (n, p, r, 1) are found in [15,29], which extend the result of Brace–Daykin[4]: w∗ (n, 1/2, r, 1) = w1/2 (G1 (n, r, 1)). To state our result let us define some families of hypergraphs as follows. G(n, r, t) = {G ⊂ 2[n] : G is r-wise t-intersecting}, Gj (n, r, t) = {G ⊂ 2[n] : G ⊂ G for some G ∼ = Gj (n, r, t)}, Gj (n, r, t). Xi (n, r, t) = G(n, r, t) − 0≤j≤i
Theorem 4.1 Let (r, t) ∈ A∗ be fixed, where A∗ is defined by (2). Then there exist positive constants γ, such that the following (i) and (ii) are true for all n ≥ r + t and p with |p − 12 | < . (i) w∗ (n, p, r, t) = wp (G1 (n, r, t)). (ii) If G ∈ X1 (n, r, t) then wp (G) < (1 − γ) w∗ (n, p, r, t). ∗ In [15] it is shown √ by construction that w (n, p, 5, 1) > wp (G1 (n, 5, 1)) for all 1/2 < p < (1 + 21)/10. Theorem 4.1 could be true for all r ≥ 5 with only exception r = 5 and t = 1, and the same extension could be expected for Theorem 2.1. The upper bound for t set by (2) in Theorem 4.1 is best possible. In fact we have wp (G2 (n, r, t)) > wp (G1 (n, r, t)) for t > 2r+1 − 3r − 1. We emphasize that Frankl has already got a special case of (i) of Theorem 4.1 in [11] (Theorem 6.4), where he proved
w∗ (n, 1/2, r, t) = w1/2 (G1 (n, r, t)) for r ≥ 5 and 1 ≤ t ≤ 2r − r − 1. Theorem 4.1 implies the following immediately. Theorem 4.2 Let (r, t) ∈ A be fixed, where A is defined by (1). Then there exists positive constant such that w(n, p, r, t) = max{wp (G0 (n, r, t)), wp (G1 (n, r, t))} holds for all n ≥ r + t and p with |p − 12 | < . We find that if (r, t) ∈ A then max{wp (G1 ), wp (G2 )} is attained by ⎧ ⎨ G (n, r, t) if 1 ≤ t ≤ 2r − r − 2, or t = 2r − r − 1 and p ≤ 1/2, 0 ⎩ G (n, r, t) if t ≥ 2r − r, or t = 2r − r − 1 and p > 1/2. 1
122
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
References [1] R. Ahlswede, L.H. Khachatrian. The complete nontrivial-intersection theorem for systems of finite sets. J. Combin. Theory (A), 76:121-138, 1996. [2] R. Ahlswede, L.H. Khachatrian. The complete intersection theorem for systems of finite sets. European J. Combin., 18:125–136, 1997. [3] R. Ahlswede, L.H. Khachatrian. The diametric theorem in Hamming spaces — Optimal anticodes. Adv. in Appl. Math., 20:429–449, 1998. [4] A. Brace, D. E. Daykin. A finite set covering theorem. Bull. Austral. Math. Soc., 5:197–202, 1971. [5] C. Bey, K. Engel. Old and new results for the weighted t-intersection problem via AK-methods. Numbers, Information and Complexity, Althofer, Ingo, Eds. et al., Dordrecht, Kluwer Academic Publishers, 45–74, 2000. [6] P. Erd˝ os, C. Ko, R. Rado. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford (2), 12:313–320, 1961. [7] I. Dinur, S. Safra. On the Hardness of Approximating Minimum Vertex-Cover. Annals of Mathematics, 162:439-485, 2005. [8] P. Frankl. On Sperner families satisfying an additional condition. J. Combin. Theory (A), 20:1–11, 1976. [9] P. Frankl. The Erd˝ os–Ko–Rado theorem is true for n = ckt. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthey, 1976), Vol. I, 365–375, Colloq. math. Soc. J´ anos Bolyai, 18, North–Holland, 1978. [10] P. Frankl. On intersecting families of finite sets. J. Combin. Theory (A), 24:141–161, 1978. [11] P. Frankl. Multiply-intersecting families. J. Combin. Theory (B), 53:195–234, 1991. [12] P. Frankl, N. Tokushige. Weighted 3-wise 2-intersecting families. J. Combin. Theory (A) 100:94–115, 2002. [13] P. Frankl, N. Tokushige. Weighted multiply intersecting families. Studia Sci. Math. Hungarica 40:287–291, 2003. [14] P. Frankl, N. Tokushige. Random walks and multiply intersecting families. J. Combin. Theory (A), 109:121-134, 2005. [15] P. Frankl, N. Tokushige. Weighted non-trivial multiply intersecting families. Combinatorica, 26 37–46, 2006.
N. Tokushige / Electronic Notes in Discrete Mathematics 28 (2007) 117–123
123
[16] E. Friedgut. On the measure of intersecting families, uniqueness and stability. preprint. [17] H.-D.O.F. Gronau. On Sperner families in which no 3-sets have an empty intersection. Acta Cybernet., 4:213–220, 1978. [18] H.-D.O.F. Gronau. On Sperner families in which no k-sets have an empty intersection. J. Combin. Theory (A), 28:54–63, 1980. [19] H.-D.O.F. Gronau. On Sperner families in which no k-sets have an empty intersection II. J. Combin. Theory (A), 30:298–316, 1981. [20] H.-D.O.F. Gronau. On Sperner families in which no k-sets have an empty intersection III. Combinatorica, 2:25–36, 1982. [21] A. J. W. Hilton, E. C. Milner. Some intersection theorems for systems of finite sets. Quart. J. Math. Oxford, 18:369–384, 1967. [22] G.O.H. Katona. Intersection theorems for systems of finite sets. Acta Math. Acad. Sci. Hung., 15:329–337, 1964. [23] E.C. Milner. A combinatorial theorem on systems of sets. J. London Math. Soc., 43:204–206, 1968. [24] D. Mubayi. A stability method in extremal set theory. preprint. [25] E. Sperner. Ein Satz u ¨ber Untermengen einer endlichen Mente. Zeitschrift, 27:544–548, 1928.
Math.
[26] N. Tokushige. Extending the Erd˝ os–Ko–Rado theorem. J. Combin. Designs, 14:52–55, 2006. [27] N. Tokushige. The maximum size of 3-wise t-intersecting families. European J. Combin, in press. [28] N. Tokushige. EKR type inequalities for 4-wise intersecting families. to appear in J. Combin. Theory (A). [29] N. Tokushige. preprint.
Brace–Daykin type inequalities for intersecting families.
[30] N. Tokushige. Intersecting families — uniform versus weighted. Ryukyu Math. J., 18:89–103, 2005. [31] R.M. Wilson. The exact bound in the Erd˝ os–Ko–Rado theorem. Combinatorica, 4:247–257, 1984.