Edges not in any monochromatic copy of a fixed graph

Abstract

For a sequence $(H_i)_{i=1}^k$ of graphs, let $\\textrm{nim}(n;H_1,\\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$. \nWhen each $H_i$ is connected and non-bipartite, we introduce a variant of Ramsey number that determines the limit of $\\textrm{nim}(n;H_1,\\ldots, H_k)/{n\\choose 2}$ as $n\\to\\infty$ and prove the corresponding stability result. Furthermore, if each $H_i$ is what we call \\emph{homomorphism-critical} (in particular if each $H_i$ is a clique), then we determine $\\textrm{nim}(n;H_1,\\ldots, H_k)$ exactly for all sufficiently large~$n$. The special case $\\textrm{nim}(n;K_3,K_3,K_3)$ of our result answers a question of Ma. \nFor bipartite graphs, we mainly concentrate on the two-colour symmetric case (i.e., when $k=2$ and $H_1=H_2$). It is trivial to see that $\\textrm{nim}(n;H,H)$ is at least $\\textrm{ex}(n,H)$, the maximum size of an $H$-free graph on $n$ vertices. Keevash and Sudakov showed that equality holds if $H$ is the $4$-cycle and $n$ is large; recently Ma extended their result to an infinite family of bipartite graphs. We provide a larger family of bipartite graphs for which $\\textrm{nim}(n;H,H)=\\textrm{ex}(n,H)$. For a general bipartite graph $H$, we show that $\\textrm{nim}(n;H,H)$ is always within a constant additive error from $\\textrm{ex}(n,H)$, i.e.,~$\\textrm{nim}(n;H,H)= \\textrm{ex}(n,H)+O_H(1)$.