Jozef Skokan Ramsey - goodness - and otherwise

A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number ∆, there is a constant r∆ such that, for any connected n-vertex graph G with maximum degree ∆, the Ramsey number R(G,G) is at most r∆n, provided n is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take r∆ = ∆. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily r∆ > 2 c∆ for some constant c > 0. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of G be at most some function β(n) = o(n), then R(G,G) ≤ (2χ(G) + 4)n ≤ (2∆ + 6)n, i.e., r∆ = 2∆ + 6 suffices. On the other hand, we show that Burr’s conjecture itself fails even for P k n , the kth power of a path Pn. Brandt showed that for any c, if ∆ is sufficiently large, there are connected nvertex graphs G with ∆(G) ≤ ∆ but R(G,K3) > cn. We show that, given ∆ and H, there are β > 0 and n0 such that, if G is a connected graph on n ≥ n0 vertices with maximum degree at most ∆ and bandwidth at most βn, then we have R(G,H) = (χ(H) − 1)(n − 1) + σ(H), where σ(H) is the smallest size of any part in any χ(H)partition of H. We also show that the same conclusion holds without any restriction on the maximum degree of G if the bandwidth of G is at most ε(H) log n/ log logn.