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 $r_Δ$ such that, for any connected $n$-vertex graph $G$ with maximum degree $Δ$, the Ramsey number $R(G,G)$ is at most $r_Δn$, provided $n$ is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take $r_Δ= Δ$. However, Graham, Rödl and Ruciński showed, by taking $G$ to be a suitable expander graph, that necessarily $r_Δ> 2^{cΔ}$ 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 $R(G,G) \le (2χ(G)+4)n\leq (2Δ+6)n$, i.e., $r_Δ= 2Δ+6$ suffices. On the other hand, we show that Burr's conjecture itself fails even for $P_n^k$, the $k$th power of a path $P_n$. Brandt showed that for any $c$, if $Δ$ is sufficiently large, there are connected $n$-vertex graphs $G$ with $Δ(G)\leqΔ$ but $R(G,K_3)>cn$. We show that, given $Δ$ and $H$, there are $β>0$ and $n_0$ such that, if $G$ is a connected graph on $n\ge n_0$ 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 $ε(H) \log n/\log\log n$.

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