Proof of a conjecture of Bollob́s and Kohayakawa on the Erdös-Stone theorem

• Authors: Yoshiyasu Ishigami
• Venue: Journal of Combinatorial Theory Series B
• Year: 2002

For any integer r ≥ 1, let a(r) be the largest constant a ≥ 0 such that if ε 0 and 0 c c0 for some small c0 = c0(r,ε) then every graph G of sufficiently large order n and at least (1 - 1/r+c)(n 2) edges contains a copy of any (r+ 1)-chromatic graph H of independence numberα(H) ≤ (a-ε) logn/log(1/c) T. Kövári et al. (1954, Colloq. Math. 2, 50-57) and B. Bollobás and P. Erdös (1973,Bull. London Math. Soc. 5, 317-321) showed that 1 ≤ a(1) ≤ 2. In an improvement to the Erdös-Stone theorem (1946), B. Bollobás et al (1976, J. London Math. Soc. (2) 12, 219-224) showed that a(r) 0 for all r and conjectured that lim infr→∞ a(r) ≠ 0. V. Chvátal and E. Szemerédi (1981, J. London Math. Soc. (2) 23, 207-214) settled it by giving a(r) ≥ 0.002 for all r. We show that, for all r, a(r) = a(1). Further we prove the conjecture of B. Bollobás and Y. Kohayakawa (1994, Combinatorica 14, 279-286). The weak form of it states that for any r ≥ 1, 0 c r, every graph G of sufficiently large order n ≥ n0(r, c) and (1-1/r+c)(n 2) edges contains any (r+ 1)-chromatic graph such that, in a proper vertex coloring, the smallest and the other color classes are of size at least βlogn\log(1/c) and βlogn\logr, respectively, for an absolute constant β 0. That is, all color classes but one are relatively large for fixed r, small c → 0, and large n → ∞. Our proof method is based on Szemerédi's Regularity Lemma.