Proof of a Conjecture of Bollobás and Kohayakawa on the Erdős–Stone Theorem

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 least1−1r+cn2 edges contains a copy of any (r+1)-chromatic graph H of independence numberα(H)⩽(a−ϵ) log nlog(1/c). T. Kővári et al. (1954, Colloq. Math.2, 50–57) and B. Bollobás and P. Erdős (1973, Bull. London Math. Soc.5, 317–321) showed that 1⩽a(1)⩽2. In an improvement to the Erdős–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, Combinatorica14, 279–286). The weak form of it states that for any r⩾1, 0<c<1/r, every graph G of sufficiently large order n⩾n0(r, c) and (1−1/r+c)(n2) 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 leastβ log nlog(1/c) and β log nlog r, 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 Szemerédiʹs Regularity Lemma.