Ramsey-goodness - and otherwise

A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number Δ, there is a constant rΔ such that, for any connected n-vertex graph G with maximum degree Δ, the Ramsey number R(G,G) is at most rΔn, provided n is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take rΔ = Δ. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily rΔ > 2cΔ 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)≤(2 χ(G)+4)n≤(2 Δ+6)n, i.e., rΔ=2 Δ+6 suffices. On the other hand, we show that Burr's conjecture itself fails even for Pn k, the kth power of a path Pn. Brandt showed that for any c, if Δ is sufficiently large, there are connected n-vertex graphs G with Δ(G)≤Δ but R(G,K3) > cn. We show that, given Δ 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 Δ 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 e{open}(H) log n=log log n. © 2013 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.