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تسجيل مستخدم جديدExact solutions to the ErdH{o}s-Rothschild problem
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Let $textbf{k} := (k_1,ldots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;textbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;textbf{k})$ to denote the maximum of $F(G;textbf{k})$ over all graphs $G$ on $n$ vertices. There are currently very few known exact (or asymptotic) results known for this problem, posed by ErdH{o}s and Rothschild in 1974. We prove some new exact results for $n to infty$: (i) A sufficient condition on $textbf{k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results. (ii) Addressing the original question of ErdH{o}s and Rothschild, in the case $textbf{k}=(3,ldots,3)$ of length $7$, the unique extremal graph is the complete balanced $8$-partite graph, with colourings coming from Hadamard matrices of order $8$. (iii) In the case $textbf{k}=(k+1,k)$, for which the sufficient condition in (i) does not hold, for $3 leq k leq 10$, the unique extremal graph is complete $k$-partite with one part of size less than $k$ and the other parts as equal in size as possible.
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Given a sequence $mathbf{k} := (k_1,ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ conta
in no clique of order $k_c$. Write $F(n;mathbf{k})$ to denote the maximum of $F(G;mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by ErdH{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. In previous work with Yilma, we constructed a finite optimisation problem whose maximum is equal to the limit of $log_2 F(n;mathbf{k})/{nchoose 2}$ as $n$ tends to infinity and proved a stability theorem for complete multipartite graphs $G$. In this paper we provide a sufficient condition on $mathbf{k}$ which guarantees a general stability theorem for any graph $G$, describing the asymptotic structure of $G$ on $n$ vertices with $F(G;mathbf{k}) = F(n;mathbf{k}) cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem. We apply our theorem to systematically recover existing stability results as well as all cases with $s=2$. The proof uses a novel version of symmetrisation on edge-coloured weighted multigraphs.
Let $mathbf{k} := (k_1,dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ cont
ain no clique of order $k_c$. Write $F(n;mathbf{k})$ to denote the maximum of $F(G;mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by ErdH{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every $mathbf{k}$ and $n$, there is a complete multipartite graph $G$ on $n$ vertices with $F(G;mathbf{k}) = F(n;mathbf{k})$. Also, for every $mathbf{k}$ we construct a finite optimisation problem whose maximum is equal to the limit of $log_2 F(n;mathbf{k})/{nchoose 2}$ as $n$ tends to infinity. Our final result is a stability theorem for complete multipartite graphs $G$, describing the asymptotic structure of such $G$ with $F(G;mathbf{k}) = F(n;mathbf{k}) cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem.
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