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- W2613342064 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~$K_n$. When each $H_i$ is connected and non-bipartite, we introduce a variant of Ramsey number that determines the limit of $textrm{nim}(n;H_1,ldots, H_k)/{nchoose 2}$ as $ntoinfty$ 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. For 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.,~$textrm{nim}(n;H,H)= textrm{ex}(n,H)+O_H(1)$." @default.
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- W2613342064 date "2017-05-04" @default.
- W2613342064 modified "2023-09-27" @default.
- W2613342064 title "Edges not in any monochromatic copy of a fixed graph" @default.
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- W2613342064 doi "https://doi.org/10.48550/arxiv.1705.01997" @default.
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