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Multicolor bipartite Ramsey numbers of Kt,s and large Kn,n
Discrete Applied Mathematics 213 (2016) 238–242
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Note
Multicolor bipartite Ramsey numbers of Kt ,s and large Kn,n ✩ Xiuwen Wang, Qizhong Lin ∗ Center for Discrete Mathematics, Fuzhou University, Fuzhou 350108, China
article
abstract
info
For any bipartite graph H, let us denote the bipartite Ramsey number brk (H ; Kn,n ) to be the minimum integer N such that any edge-coloring of the complete bipartite graph KN ,N by k + 1 colors contains a monochromatic copy of H in some color i for 1 ≤ i ≤ k, or a monochromatic copy of Kn,n in the last color. In this note, it is shown that for any fixed integers t ≥ 2 and s ≥ (t − 1)! + 1, there exists a constant c = c (t ) > 0 such n log log n t for sufficiently large n; and for k ≥ 3, brk (Kt ,s ; Kn,n ) = that br2 (Kt ,s ; Kn,n ) ≥ c 2
Article history: Received 6 February 2015 Received in revised form 16 March 2016 Accepted 2 May 2016 Available online 28 May 2016 Keywords: Bipartite Ramsey number Turán number Asymptotic bound
log n
Θ
nt logt n
.
© 2016 Elsevier B.V. All rights reserved.
1. Introduction Let H1 , H2 , . . . , Hk+1 be a sequence of bipartite graphs. The bipartite Ramsey number br (H1 , H2 , . . . , Hk+1 ) is defined to be the minimum integer N such that any edge-coloring of the complete bipartite graph KN ,N by k + 1 colors contains a monochromatic copy of Hi in some color i. We shall denote br (H1 , . . . , Hk , Hk+1 ) by brk (H ; Hk+1 ) if Hi = H for i = 1, . . . , k. The two-color case br (H1 , H2 ) has attracted most attention. Thomason [17] proved that br (Kt ,n , Kt ,n ) ≤ 2t (n − 1) + 1 for any n ≥ t ≥ 1. The special cases for t = 2 and t = 3 are due to Beineke and Schwenk [3], and Irving [8], respectively. It was shown that br (Kn,n , Kn,n ) ≤ (1 + o(1))2n+1 log2 n by Conlon [6] as n → ∞, and br (Kt ,n , Kt ,n ) = (1 + o(1))2t n for fixed t by Li, Tang and Zang [13]. Caro and Rousseau [5] proved that for fixed t ≥ 2, c1
n (t +1)/2 log n
< br (Kt ,t , Kn,n ) < c2
n t log n
,
where ci = ci (t ) > 0 are constants, and here and henceforth logarithmic function has natural base e. Lin and Li [14] showed that the order of magnitude of br (Kt ,n , Kn,n ) is nt +1 /(log n)t , but the order of magnitude of br (Kt ,t , Kn,n ), which is a more interesting problem, seems to be very hard to obtain. For k ≥ 3, the situation is much more difficult. An (N , d, λ)-graph is a d-regular graph G of order N, where λ is the second largest eigenvalue in absolute value of the adjacency matrix of G. Here an (N , d, λ)-graph may contain loops (at most one loop per vertex), where loops contribute one to the degree. By generalizing the method by Alon and Rödl [1] that estimating the number of Kn in an (N , d, λ)-graph. Lin and Li [15] estimated the number of Kn,n in an (N , d, λ)-graph, from which the authors obtained that c1
n log log n 2 log2 n
≤ br2 (K2,2 ; Kn,n ) ≤ c2
2 n , log n
and for k ≥ 3, brk (K2,2 ; Kn,n ) = Θ (n2 / log2 n).
In this note, we will mainly consider the bipartite Ramsey number brk (Kt ,s ; Kn,n ) for k ≥ 2, t ≥ 2 and s ≥ (t − 1)! + 1 fixed and n sufficiently large.
✩ Supported in part by CSC (grant no. 201405566002) and NSFC.
∗
Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (Q. Lin).
http://dx.doi.org/10.1016/j.dam.2016.05.002 0166-218X/© 2016 Elsevier B.V. All rights reserved.
X. Wang, Q. Lin / Discrete Applied Mathematics 213 (2016) 238–242
239
Theorem 1. Let t ≥ 2 and k ≥ 3 be fixed, and let s ≥ (t − 1)! + 1. Then, there exists a constant c = c (t ) > 0 such that
(1) br2 (Kt ,s ; Kn,n ) ≥ c
n log log n t
(2) brk (Kt ,s ; Kn,n ) = Θ
log2 n
,
nt logt n
for sufficiently large n. 2. Proofs of the main result Let H1 , H2 , . . . , Hk+1 be a sequence of k + 1 finite, undirected, simple bipartite graphs. The multicolor Ramsey number r (H1 , H2 , . . . , Hk+1 ) is defined to be the smallest integer N such that in any edge coloring of complete graph KN by colors 1, 2, . . . , k + 1, there exists a monochromatic copy of Hi in color i for some 1 ≤ i ≤ k + 1. We will also denote r (H1 , H2 , . . . , Hk+1 ) by rk (H ; Hk+1 ) if Hi = H for 1 ≤ i ≤ k. Clearly, r (H1 , H2 , . . . , Hk+1 ) ≤ 2 br (H1 , H2 , . . . , Hk+1 ),
(1)
which can be seen from the fact that an edge-coloring of K2N induces an edge-coloring of KN ,N . Let us establish several useful lemmas at first. The following result, which is shown in [15], implies that any (N , d, λ)graph G with the property that the complement graph G does not contain too many Kn,n if n is much larger than Nd log N. Lemma 1 ([15, Theorem 3 & Lemma 3]). Let G be an (N , d, λ)-graph which is H-free, and let M denote the number of Kn,n in G. If n ≥ 4N log N, then d M ≤
N 4N log 2eλN 2n d
ed2 n
4λN log N
dn
Moreover, for an integer k ≥ 2, if M k ≤
.
N −n n
2(k−1)
, then rk (H ; Kn,n ) > N.
The projective norm graphs G(p, t ) have been constructed by Alon, Rónyai and Szabó [2], modifying an earlier construction given by Kollar, Rónyai and Szabó [9]. The construction is as follows. Let t ≥ 2 be an integer, let p be a prime, let GF (p)∗ denote the multiplicative group of the finite field with p elements, and let GF (pt −1 ) denote the field with pt −1 elements. The set of vertices of the graph G = G(p, t ) is the set V = GF (pt −1 ) × GF (p)∗ . Two distinct vertices (X , a) and (t −2)
(Y , b) ∈ V are adjacent if and only if R(X + Y ) = ab, where the norm N is understood over GF (p), i.e., R(Z ) = Z 1+p+···+p The following property of G(p, t ) was proved in [2] by using some tools from algebraic geometry described in [9].
.
Lemma 2 ([2, Theorem 5]). The graph G(p, t ) contains no copy of Kt ,(t −1)!+1 . In order to estimate the number of Kn,n in G(p, t ), we also need the eigenvalues of G(p, t ). The following result is obtained independently by Szabó [16], and Alon and Rödl [1]. Lemma 3 ([16, Theorem 1]; [1, Lemma 3.6]). Let G = G(p, t ) be defined as above. Then G is an (N , d, λ)-graph with N = pt − pt −1 , d = pt −1 − 1 and λ = p(t −1)/2 . Now, we are ready to give a proof for Theorem 1. First, we shall show that part (1) of Theorem 1 holds. Lemma 4. Let t ≥ 2 and s ≥ (t − 1)! + 1 be fixed integers. Then, there exists a constant c = c (t ) > 0 such that r 2 ( K t ,s ; K n ,n ) ≥ c
n log log n t log2 n
for sufficiently large n.
t
Proof. Let N1 = c1 n log log n/ log2 n , where c1 = 1/t 2t . Let p be the maximum prime such that pt − pt −1 < N1 , and let G = G(p, t ) be the Kt ,s -free (N , d, λ)-graph of order N = pt − pt −1 . From the fact that there exists a prime p such that
1/t
1/t
m < p ≤ 2m for any integer m ≥ 1, we have ⌊N1 /2⌋ ≤ p ≤ ⌊N1 ⌋, and hence N1 /22t +1 ≤ pt /2 ≤ pt − pt −1 = N < N1 .
t
Write N = c n log log n/ log2 n , where c is a constant with 1/(22t +1 t 2t ) < c < 1/t 2t . A simple calculation implies that
n=Θ
N 1/t log2 N log log N
ed2 4λ
. Hence, we have n ≥
< N 3(t −1)/(2t ) ,
2eλ d
<
4N log N d
by noticing that d ∼ N 1−1/t . For this graph G, λ ∼ d1/2 , and hence we have
6 N (t −1)/(2t )
,
4N log N d
< 4N 1/t log N .
240
X. Wang, Q. Lin / Discrete Applied Mathematics 213 (2016) 238–242
Let M denote the number of Kn,n in G. From Lemma 1, it remains only to check that M <
N −n n
since G contains no Kt ,s . In
fact, Lemma 1 implies that the number M of Kn,n in G satisfies that
M ≤
nN (t −3)/(2t )
6N (t +1)/(2t )
log N
As n = Θ
2n
.
n
(2)
N 1/t log2 N
, we have that there exist some constants ai = ai (t ) > 0 such that
log log N
M ≤
4N 1/t log N
a1 N (t −1)/(2t ) log N
a2 N (t −1)/(2t ) log log N
N 1/t log2 N t 2 c 1/t log log N
a1 N (t −1)/(2t ) log N
2n
log2 N
log log N
Note also that n ≥
4N 1/t log N
.
t
since N = c n log log n/ log2 n , we have that
log N 4t 2 c 1/logt log N
a2 N (t −1)/(2t ) log log N
2 ≤
2
log log N
log N
4N 1/t log N n
<
a3 N (t −1)/t log log2 N
(log N )4−2t (t −1)c 1/t
4t 2 c 1/t log log N log N
and M 1/n is at most
.
However,
N −n
1/n =
n
(1 − o(1))eN n
≥
a4 N 1−1/t log log N log2 N
,
and the desired inequality follows for large N as 4 − 2t (t − 1)c 1/t > 2.
Proof of (1) of Theorem 1. Since r2 (Kt ,s ; Kn,n ) ≤ 2br2 (Kt ,s ; Kn,n ), the lower bound on br2 (Kt ,s ; Kn,n ) follows from Lemma 4 immediately. Remark 1. The proof of Theorem 1 will also imply that result by Lenz and Mubayi [12], namely for any fixed s ≥ 2 and sufficiently large n, the lower bound of r2 (K2,s ; Kn ) can be improved by a factor of log log2 n. For the lower bound of (2) of Theorem 1, it suffices to prove the following lemma. Lemma 5. Let t ≥ 2 and s ≥ (t − 1)! + 1 be fixed integers. We have that there exists a constant c = c (t ) > 0 such that r 3 ( K t ,s ; K n ,n ) ≥ c
n t log n
for sufficiently large n.
t
Proof. Let N1 = c1 n/ log n , where c1 = 1/(6t )t . Similar to that in Lemma 4, let G = G(p, t ) be the Kt ,s -free (N , d, λ)-
t
graph of order N = pt − pt −1 < N1 for the maximum prime p. Write N = c n/ log n , where c is a constant with 1/(24t )t < c < 1/(6t )t .
Note that n ≥
M ≤
N 1/t log N tc 1/t
nN (t −3)/(2t )
>
4N log N d
for large N, Lemma 1 implies that the number M of Kn,n in G satisfies that
4N 1/t log N
6N (t +1)/(2t )
log N
≤ a1 N
2n
n
4N 1/t log N a N (t −1)/(2t ) 2n 2 (t −1)/(2t ) log N
,
where ai = ai (t ) > 0 are constants. Therefore,
M 3/n ≤ a1 N (t −1)/(2t )
≤
a3 N
12tc 1/t a N (t −1)/(2t ) 6 2 log N
6(t −1)c 1/t +3(t −1)/t
log6 N
.
However,
N −n n
4/n =
(1 − o(1))eN 4 n
≥
a4 N 4(t −1)/t log4 N
,
and the desired inequality follows for large N as c < 1/(6t )t .
X. Wang, Q. Lin / Discrete Applied Mathematics 213 (2016) 238–242
Proof of lower bound of (2) of Theorem 1. The lower bound of brk (Kt ,s ; Kn,n ) for k immediately.
241
≥
3 follows from Lemma 5
In the following, we shall focus on the proof of the upper bound of (2) of Theorem 1. Given a graph H, the Turán number ex(N ; H ) is the maximum number of edges in an H-free graph on N vertices. A famous result of Kövári, Sós, and Turán [10] gave that 1
(s − 1)1/t N 1−1/t (N − t + 1) + (t − 1)N , (3) 2 1 where t ≤ s. For large N, the upper bound above was improved by Füredi [7] as 2 (s − t + 1)1/t N 2−1/t + o(N 2−1/t ). Given a bipartite graph H, the Zarankiewicz number z (N ; H ) is the maximum number of edges in an H-free bipartite graph G ⊆ KN ,N . In [10], it also gave that ex(N ; Kt ,s ) ≤
z (N ; Kt ,s ) ≤ (s − 1)1/t N 1−1/t (N − t + 1) + (t − 1)N .
(4)
Lemma 6. Let H be a bipartite graph with ex(N ; H ) = O(N 2−1/t ). Then for any fixed k ≥ 1, there is a constant c = c (H , k, t ) > 0 such that n t brk (H ; Kn,n ) ≤ c log n for sufficient large n. Proof. Set N = c (n/ log n)t , where c = c (k, t ) is a positive constant which will be chosen in the following way. We shall prove that brk (H ; Kn,n ) ≤ N. From the definition of the Zarankiewicz number, it suffices to show that k · ex(2N ; H ) + z (N ; Kn,n ) < N 2 . From the assumption that there exists some constant c1 = c1 (t ) > 0 such that ex(2N ; H ) ≤ c1 N 2−1/t , which implies that ex(2N ; H ) N2
<
c1 N 1/t
c1 log n
=
c 1/t n
.
Note that ex − 1 = x + o(x) as x → 0, hence for sufficiently large n,
n 1/n N
− 1 = e(log n−log N )/n − 1 = e[(1−t ) log n+t log log n−log c ]/n − 1 log log n log n +O . = −(t − 1) n
n
Consequently, from (4), we have that z ( N ; K n ,n )
<
n 1n
1−
n − 1
n
+ N log log n = 1 − (t − 1) +O . n n Therefore, k · ex(2N ; H ) + z (N ; Kn,n ) /N 2 is bounded from above by log log n kc1 log n 1 − t − 1 − 1/t +O , N2
N
c
N log n
n
n
which is less than 1 for large n if we choose c > (kc1 )t . This completes the proof of Lemma 6. Proof of upper bound of (2) of Theorem 1. Since ex(N ; Kt ,s ) ≤ O(N Lemma 6.
2−1/t
), the upper bound of brk (Kt ,s ; Kn,n ) follows from
By slightly changing the proof of Theorem 1, we can obtain a general result as follows. Theorem 2. Let H be a fixedbipartitegraph with ex(N ; H ) = O(N 2−1/t ), and suppose that there exists an (N , d, λ)-graph which is H-free satisfying d = Θ N (t −1)/t and λ = Θ N (t −1)/(2t ) . Then for any fixed t ≥ 2 and k ≥ 3, there exists a constant c = c (t , k) > 0 such that
(1) br2 (H ; Kn,n ) ≥ c
n log log n t
(2) brk (H ; Kn,n ) = Θ for sufficiently large n.
log2 n
nt logt n
,
242
X. Wang, Q. Lin / Discrete Applied Mathematics 213 (2016) 238–242
3. Concluding remarks Throughout the proof of Theorem 1, it is crucial to find a suitable (N , d, λ)-graph that is H-free, where H is bipartite. Let Cm be a cycle of length m. It is known that ex(N ; C6 ) = O(N 2−2/3 ) by Bondy and Simonovits [4]. Moreover, let q be an odd power of 2, there exists a polarity graph G of order N = q3 + q2 + q + 1 which is a (q + 1)-regular graph,√see Lazebnik, √Ustimenko and Woldar [11] for more details. This graph contains no C6 and all the eigenvalues of G are q + 1, 2q, 0 and − 2q, see [1, pp. 138]. Hence G is an (N , d, λ)-graph with d ∼ N 1/3 and λ ∼ N 1/6 . Therefore, Theorem 2 implies that br2 (C6 ; Kn,n ) = Ω
n3/2 log log3/2 n log3 n
and brk (C6 ; Kn,n ) = Θ
n3/2 log3/2 n 2−4/5
for k ≥ 3.
Another example is for C10 . Note that ex(N ; C10 ) = O(N ), see [4]. Also, for every q which is an odd power of 3, there exists a polarity graph G of order N = q5 + q4 + q3 + q2 + q + 1 which √ is a (q + 1)-regular graph, see [11] for more √ details. This graph contains no C10 and all the eigenvalues of G are q + 1, ± 3q and ± q, see [1, pp. 138]. Hence G is an
(N , d, λ)-graph with d ∼ N 1/5 and λ ∼ N 1/10 . Therefore, Theorem 2 implies that br2 (C10 ; Kn,n ) = Ω 5/4 brk (C10 ; Kn,n ) = Θ n5/4 for k ≥ 3. log n
n5/4 log log5/4 n log5/2 n
and
Acknowledgments We are grateful to the anonymous referees for the invaluable comments and suggestions, which have improved the presentation of the manuscript greatly. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
N. Alon, V. Rödl, Sharp bounds for some multicolor Ramsey numbers, Combinatorica 25 (2005) 125–141. N. Alon, L. Rónyai, T. Szabó, Norm-graphs: variations and applications, J. Combin. Theory Ser. B 76 (1999) 280–290. L. Beineke, A. Schwenk, On a bipartite form of the Ramsey problem, in: Proc. 5th British Combin. Conf. 1975, Utilitas Math, Winnipeg, 1976, pp. 17–22. A.J. Bondy, M. Simonovits, Cycles of even lengths in graphs, J. Combin. Theory Ser. B 16 (1974) 97–105. Y. Caro, C. Rousseau, Asymptotic bounds for bipartite Ramsey numbers, Electron. J. Combin. 8 (1) (2001) R17. D. Conlon, A new upper bound for the bipartite Ramsey problem, J. Graph Theory 58 (2008) 351–356. Z. Füredi, An upper bound on Zarankiewicz’ problem, Combin. Probab. Comput. 5 (1996) 29–33. R. Irving, A bipartite Ramsey problem and the Zarankiewicz number, Glasg. Math. J. 19 (1978) 13–26. J. Kollar, L. Rónyai, T. Szábo, Norm-graphs and bipartite Turán numbers, Combinatorica 16 (1996) 399–406. T. Kövári, T. Sós, P. Turán, On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954) 50–57. F. Lazebnik, V.A. Ustimenko, A.J. Woldar, Polarities and 2k-cycle-free graphs, Discrete Math. 197/198 (1999) 503–513. J. Lenz, D. Mubayi, Multicolor Ramsey number for complete bipartive versus complete graph, J. Graph Theory (2012) 20–38. Y. Li, X. Tang, W. Zang, Ramsey functions involving Km,n with n large, Discrete Math. 300 (2005) 120–128. Q. Lin, Y. Li, Bipartite Ramsey numbers involving large Kn,n , European J. Combin. 30 (2009) 923–928. Q. Lin, Y. Li, Multicolor bipartite Ramsey number of C4 and large Kn,n , J. Graph Theory 67 (2011) 47–54. T. Szabó, On the spectrum of projective norm-graphs, Inform. Process. Lett. 86 (2003) 71–74. A. Thomason, On finite Ramsey numbers, European J. Combin. 3 (1982) 263–273.
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Note
Multicolor bipartite Ramsey numbers of Kt ,s and large Kn,n ✩ Xiuwen Wang, Qizhong Lin ∗ Center for Discrete Mathematics, Fuzhou University, Fuzhou 350108, China
article
abstract
info
For any bipartite graph H, let us denote the bipartite Ramsey number brk (H ; Kn,n ) to be the minimum integer N such that any edge-coloring of the complete bipartite graph KN ,N by k + 1 colors contains a monochromatic copy of H in some color i for 1 ≤ i ≤ k, or a monochromatic copy of Kn,n in the last color. In this note, it is shown that for any fixed integers t ≥ 2 and s ≥ (t − 1)! + 1, there exists a constant c = c (t ) > 0 such n log log n t for sufficiently large n; and for k ≥ 3, brk (Kt ,s ; Kn,n ) = that br2 (Kt ,s ; Kn,n ) ≥ c 2
Article history: Received 6 February 2015 Received in revised form 16 March 2016 Accepted 2 May 2016 Available online 28 May 2016 Keywords: Bipartite Ramsey number Turán number Asymptotic bound
log n
Θ
nt logt n
.
© 2016 Elsevier B.V. All rights reserved.
1. Introduction Let H1 , H2 , . . . , Hk+1 be a sequence of bipartite graphs. The bipartite Ramsey number br (H1 , H2 , . . . , Hk+1 ) is defined to be the minimum integer N such that any edge-coloring of the complete bipartite graph KN ,N by k + 1 colors contains a monochromatic copy of Hi in some color i. We shall denote br (H1 , . . . , Hk , Hk+1 ) by brk (H ; Hk+1 ) if Hi = H for i = 1, . . . , k. The two-color case br (H1 , H2 ) has attracted most attention. Thomason [17] proved that br (Kt ,n , Kt ,n ) ≤ 2t (n − 1) + 1 for any n ≥ t ≥ 1. The special cases for t = 2 and t = 3 are due to Beineke and Schwenk [3], and Irving [8], respectively. It was shown that br (Kn,n , Kn,n ) ≤ (1 + o(1))2n+1 log2 n by Conlon [6] as n → ∞, and br (Kt ,n , Kt ,n ) = (1 + o(1))2t n for fixed t by Li, Tang and Zang [13]. Caro and Rousseau [5] proved that for fixed t ≥ 2, c1
n (t +1)/2 log n
< br (Kt ,t , Kn,n ) < c2
n t log n
,
where ci = ci (t ) > 0 are constants, and here and henceforth logarithmic function has natural base e. Lin and Li [14] showed that the order of magnitude of br (Kt ,n , Kn,n ) is nt +1 /(log n)t , but the order of magnitude of br (Kt ,t , Kn,n ), which is a more interesting problem, seems to be very hard to obtain. For k ≥ 3, the situation is much more difficult. An (N , d, λ)-graph is a d-regular graph G of order N, where λ is the second largest eigenvalue in absolute value of the adjacency matrix of G. Here an (N , d, λ)-graph may contain loops (at most one loop per vertex), where loops contribute one to the degree. By generalizing the method by Alon and Rödl [1] that estimating the number of Kn in an (N , d, λ)-graph. Lin and Li [15] estimated the number of Kn,n in an (N , d, λ)-graph, from which the authors obtained that c1
n log log n 2 log2 n
≤ br2 (K2,2 ; Kn,n ) ≤ c2
2 n , log n
and for k ≥ 3, brk (K2,2 ; Kn,n ) = Θ (n2 / log2 n).
In this note, we will mainly consider the bipartite Ramsey number brk (Kt ,s ; Kn,n ) for k ≥ 2, t ≥ 2 and s ≥ (t − 1)! + 1 fixed and n sufficiently large.
✩ Supported in part by CSC (grant no. 201405566002) and NSFC.
∗
Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (Q. Lin).
http://dx.doi.org/10.1016/j.dam.2016.05.002 0166-218X/© 2016 Elsevier B.V. All rights reserved.
X. Wang, Q. Lin / Discrete Applied Mathematics 213 (2016) 238–242
239
Theorem 1. Let t ≥ 2 and k ≥ 3 be fixed, and let s ≥ (t − 1)! + 1. Then, there exists a constant c = c (t ) > 0 such that
(1) br2 (Kt ,s ; Kn,n ) ≥ c
n log log n t
(2) brk (Kt ,s ; Kn,n ) = Θ
log2 n
,
nt logt n
for sufficiently large n. 2. Proofs of the main result Let H1 , H2 , . . . , Hk+1 be a sequence of k + 1 finite, undirected, simple bipartite graphs. The multicolor Ramsey number r (H1 , H2 , . . . , Hk+1 ) is defined to be the smallest integer N such that in any edge coloring of complete graph KN by colors 1, 2, . . . , k + 1, there exists a monochromatic copy of Hi in color i for some 1 ≤ i ≤ k + 1. We will also denote r (H1 , H2 , . . . , Hk+1 ) by rk (H ; Hk+1 ) if Hi = H for 1 ≤ i ≤ k. Clearly, r (H1 , H2 , . . . , Hk+1 ) ≤ 2 br (H1 , H2 , . . . , Hk+1 ),
(1)
which can be seen from the fact that an edge-coloring of K2N induces an edge-coloring of KN ,N . Let us establish several useful lemmas at first. The following result, which is shown in [15], implies that any (N , d, λ)graph G with the property that the complement graph G does not contain too many Kn,n if n is much larger than Nd log N. Lemma 1 ([15, Theorem 3 & Lemma 3]). Let G be an (N , d, λ)-graph which is H-free, and let M denote the number of Kn,n in G. If n ≥ 4N log N, then d M ≤
N 4N log 2eλN 2n d
ed2 n
4λN log N
dn
Moreover, for an integer k ≥ 2, if M k ≤
.
N −n n
2(k−1)
, then rk (H ; Kn,n ) > N.
The projective norm graphs G(p, t ) have been constructed by Alon, Rónyai and Szabó [2], modifying an earlier construction given by Kollar, Rónyai and Szabó [9]. The construction is as follows. Let t ≥ 2 be an integer, let p be a prime, let GF (p)∗ denote the multiplicative group of the finite field with p elements, and let GF (pt −1 ) denote the field with pt −1 elements. The set of vertices of the graph G = G(p, t ) is the set V = GF (pt −1 ) × GF (p)∗ . Two distinct vertices (X , a) and (t −2)
(Y , b) ∈ V are adjacent if and only if R(X + Y ) = ab, where the norm N is understood over GF (p), i.e., R(Z ) = Z 1+p+···+p The following property of G(p, t ) was proved in [2] by using some tools from algebraic geometry described in [9].
.
Lemma 2 ([2, Theorem 5]). The graph G(p, t ) contains no copy of Kt ,(t −1)!+1 . In order to estimate the number of Kn,n in G(p, t ), we also need the eigenvalues of G(p, t ). The following result is obtained independently by Szabó [16], and Alon and Rödl [1]. Lemma 3 ([16, Theorem 1]; [1, Lemma 3.6]). Let G = G(p, t ) be defined as above. Then G is an (N , d, λ)-graph with N = pt − pt −1 , d = pt −1 − 1 and λ = p(t −1)/2 . Now, we are ready to give a proof for Theorem 1. First, we shall show that part (1) of Theorem 1 holds. Lemma 4. Let t ≥ 2 and s ≥ (t − 1)! + 1 be fixed integers. Then, there exists a constant c = c (t ) > 0 such that r 2 ( K t ,s ; K n ,n ) ≥ c
n log log n t log2 n
for sufficiently large n.
t
Proof. Let N1 = c1 n log log n/ log2 n , where c1 = 1/t 2t . Let p be the maximum prime such that pt − pt −1 < N1 , and let G = G(p, t ) be the Kt ,s -free (N , d, λ)-graph of order N = pt − pt −1 . From the fact that there exists a prime p such that
1/t
1/t
m < p ≤ 2m for any integer m ≥ 1, we have ⌊N1 /2⌋ ≤ p ≤ ⌊N1 ⌋, and hence N1 /22t +1 ≤ pt /2 ≤ pt − pt −1 = N < N1 .
t
Write N = c n log log n/ log2 n , where c is a constant with 1/(22t +1 t 2t ) < c < 1/t 2t . A simple calculation implies that
n=Θ
N 1/t log2 N log log N
ed2 4λ
. Hence, we have n ≥
< N 3(t −1)/(2t ) ,
2eλ d
<
4N log N d
by noticing that d ∼ N 1−1/t . For this graph G, λ ∼ d1/2 , and hence we have
6 N (t −1)/(2t )
,
4N log N d
< 4N 1/t log N .
240
X. Wang, Q. Lin / Discrete Applied Mathematics 213 (2016) 238–242
Let M denote the number of Kn,n in G. From Lemma 1, it remains only to check that M <
N −n n
since G contains no Kt ,s . In
fact, Lemma 1 implies that the number M of Kn,n in G satisfies that
M ≤
nN (t −3)/(2t )
6N (t +1)/(2t )
log N
As n = Θ
2n
.
n
(2)
N 1/t log2 N
, we have that there exist some constants ai = ai (t ) > 0 such that
log log N
M ≤
4N 1/t log N
a1 N (t −1)/(2t ) log N
a2 N (t −1)/(2t ) log log N
N 1/t log2 N t 2 c 1/t log log N
a1 N (t −1)/(2t ) log N
2n
log2 N
log log N
Note also that n ≥
4N 1/t log N
.
t
since N = c n log log n/ log2 n , we have that
log N 4t 2 c 1/logt log N
a2 N (t −1)/(2t ) log log N
2 ≤
2
log log N
log N
4N 1/t log N n
<
a3 N (t −1)/t log log2 N
(log N )4−2t (t −1)c 1/t
4t 2 c 1/t log log N log N
and M 1/n is at most
.
However,
N −n
1/n =
n
(1 − o(1))eN n
≥
a4 N 1−1/t log log N log2 N
,
and the desired inequality follows for large N as 4 − 2t (t − 1)c 1/t > 2.
Proof of (1) of Theorem 1. Since r2 (Kt ,s ; Kn,n ) ≤ 2br2 (Kt ,s ; Kn,n ), the lower bound on br2 (Kt ,s ; Kn,n ) follows from Lemma 4 immediately. Remark 1. The proof of Theorem 1 will also imply that result by Lenz and Mubayi [12], namely for any fixed s ≥ 2 and sufficiently large n, the lower bound of r2 (K2,s ; Kn ) can be improved by a factor of log log2 n. For the lower bound of (2) of Theorem 1, it suffices to prove the following lemma. Lemma 5. Let t ≥ 2 and s ≥ (t − 1)! + 1 be fixed integers. We have that there exists a constant c = c (t ) > 0 such that r 3 ( K t ,s ; K n ,n ) ≥ c
n t log n
for sufficiently large n.
t
Proof. Let N1 = c1 n/ log n , where c1 = 1/(6t )t . Similar to that in Lemma 4, let G = G(p, t ) be the Kt ,s -free (N , d, λ)-
t
graph of order N = pt − pt −1 < N1 for the maximum prime p. Write N = c n/ log n , where c is a constant with 1/(24t )t < c < 1/(6t )t .
Note that n ≥
M ≤
N 1/t log N tc 1/t
nN (t −3)/(2t )
>
4N log N d
for large N, Lemma 1 implies that the number M of Kn,n in G satisfies that
4N 1/t log N
6N (t +1)/(2t )
log N
≤ a1 N
2n
n
4N 1/t log N a N (t −1)/(2t ) 2n 2 (t −1)/(2t ) log N
,
where ai = ai (t ) > 0 are constants. Therefore,
M 3/n ≤ a1 N (t −1)/(2t )
≤
a3 N
12tc 1/t a N (t −1)/(2t ) 6 2 log N
6(t −1)c 1/t +3(t −1)/t
log6 N
.
However,
N −n n
4/n =
(1 − o(1))eN 4 n
≥
a4 N 4(t −1)/t log4 N
,
and the desired inequality follows for large N as c < 1/(6t )t .
X. Wang, Q. Lin / Discrete Applied Mathematics 213 (2016) 238–242
Proof of lower bound of (2) of Theorem 1. The lower bound of brk (Kt ,s ; Kn,n ) for k immediately.
241
≥
3 follows from Lemma 5
In the following, we shall focus on the proof of the upper bound of (2) of Theorem 1. Given a graph H, the Turán number ex(N ; H ) is the maximum number of edges in an H-free graph on N vertices. A famous result of Kövári, Sós, and Turán [10] gave that 1
(s − 1)1/t N 1−1/t (N − t + 1) + (t − 1)N , (3) 2 1 where t ≤ s. For large N, the upper bound above was improved by Füredi [7] as 2 (s − t + 1)1/t N 2−1/t + o(N 2−1/t ). Given a bipartite graph H, the Zarankiewicz number z (N ; H ) is the maximum number of edges in an H-free bipartite graph G ⊆ KN ,N . In [10], it also gave that ex(N ; Kt ,s ) ≤
z (N ; Kt ,s ) ≤ (s − 1)1/t N 1−1/t (N − t + 1) + (t − 1)N .
(4)
Lemma 6. Let H be a bipartite graph with ex(N ; H ) = O(N 2−1/t ). Then for any fixed k ≥ 1, there is a constant c = c (H , k, t ) > 0 such that n t brk (H ; Kn,n ) ≤ c log n for sufficient large n. Proof. Set N = c (n/ log n)t , where c = c (k, t ) is a positive constant which will be chosen in the following way. We shall prove that brk (H ; Kn,n ) ≤ N. From the definition of the Zarankiewicz number, it suffices to show that k · ex(2N ; H ) + z (N ; Kn,n ) < N 2 . From the assumption that there exists some constant c1 = c1 (t ) > 0 such that ex(2N ; H ) ≤ c1 N 2−1/t , which implies that ex(2N ; H ) N2
<
c1 N 1/t
c1 log n
=
c 1/t n
.
Note that ex − 1 = x + o(x) as x → 0, hence for sufficiently large n,
n 1/n N
− 1 = e(log n−log N )/n − 1 = e[(1−t ) log n+t log log n−log c ]/n − 1 log log n log n +O . = −(t − 1) n
n
Consequently, from (4), we have that z ( N ; K n ,n )
<
n 1n
1−
n − 1
n
+ N log log n = 1 − (t − 1) +O . n n Therefore, k · ex(2N ; H ) + z (N ; Kn,n ) /N 2 is bounded from above by log log n kc1 log n 1 − t − 1 − 1/t +O , N2
N
c
N log n
n
n
which is less than 1 for large n if we choose c > (kc1 )t . This completes the proof of Lemma 6. Proof of upper bound of (2) of Theorem 1. Since ex(N ; Kt ,s ) ≤ O(N Lemma 6.
2−1/t
), the upper bound of brk (Kt ,s ; Kn,n ) follows from
By slightly changing the proof of Theorem 1, we can obtain a general result as follows. Theorem 2. Let H be a fixedbipartitegraph with ex(N ; H ) = O(N 2−1/t ), and suppose that there exists an (N , d, λ)-graph which is H-free satisfying d = Θ N (t −1)/t and λ = Θ N (t −1)/(2t ) . Then for any fixed t ≥ 2 and k ≥ 3, there exists a constant c = c (t , k) > 0 such that
(1) br2 (H ; Kn,n ) ≥ c
n log log n t
(2) brk (H ; Kn,n ) = Θ for sufficiently large n.
log2 n
nt logt n
,
242
X. Wang, Q. Lin / Discrete Applied Mathematics 213 (2016) 238–242
3. Concluding remarks Throughout the proof of Theorem 1, it is crucial to find a suitable (N , d, λ)-graph that is H-free, where H is bipartite. Let Cm be a cycle of length m. It is known that ex(N ; C6 ) = O(N 2−2/3 ) by Bondy and Simonovits [4]. Moreover, let q be an odd power of 2, there exists a polarity graph G of order N = q3 + q2 + q + 1 which is a (q + 1)-regular graph,√see Lazebnik, √Ustimenko and Woldar [11] for more details. This graph contains no C6 and all the eigenvalues of G are q + 1, 2q, 0 and − 2q, see [1, pp. 138]. Hence G is an (N , d, λ)-graph with d ∼ N 1/3 and λ ∼ N 1/6 . Therefore, Theorem 2 implies that br2 (C6 ; Kn,n ) = Ω
n3/2 log log3/2 n log3 n
and brk (C6 ; Kn,n ) = Θ
n3/2 log3/2 n 2−4/5
for k ≥ 3.
Another example is for C10 . Note that ex(N ; C10 ) = O(N ), see [4]. Also, for every q which is an odd power of 3, there exists a polarity graph G of order N = q5 + q4 + q3 + q2 + q + 1 which √ is a (q + 1)-regular graph, see [11] for more √ details. This graph contains no C10 and all the eigenvalues of G are q + 1, ± 3q and ± q, see [1, pp. 138]. Hence G is an
(N , d, λ)-graph with d ∼ N 1/5 and λ ∼ N 1/10 . Therefore, Theorem 2 implies that br2 (C10 ; Kn,n ) = Ω 5/4 brk (C10 ; Kn,n ) = Θ n5/4 for k ≥ 3. log n
n5/4 log log5/4 n log5/2 n
and
Acknowledgments We are grateful to the anonymous referees for the invaluable comments and suggestions, which have improved the presentation of the manuscript greatly. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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