1. Edges not in any monochromatic copy of a fixed graph
- Author
-
Maryam Sharifzadeh, Oleg Pikhurko, and Hong Liu
- Subjects
Additive error ,010102 general mathematics ,0102 computer and information sciences ,Symmetric case ,Stability result ,01 natural sciences ,Graph ,Theoretical Computer Science ,Combinatorics ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,FOS: Mathematics ,Bipartite graph ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Maximum size ,Combinatorics (math.CO) ,Monochromatic color ,Ramsey's theorem ,0101 mathematics ,QA ,Mathematics - 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 $\textrm{nim}(n;H_1,\ldots, H_k)/{n\choose 2}$ 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. 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.,~$\textrm{nim}(n;H,H)= \textrm{ex}(n,H)+O_H(1)$., 24 pages, 3 figures
- Published
- 2019