For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In his 1977 formulation of the problem, Erd\H{o}s discussed the small values of $k$ and left the cases $k=4$ and $k=5$ as the next natural open questions. For $k=4$ he asked whether $f(n,4)=2n-4$, with the upper bound witnessed by $K_{2,n-2}$; the neighbouring $k=5$ question was recently settled exactly by Fang and Hou. We prove that every graph $G$ of order $n$ and size at most $2n-10n^{2/3}-7$ has a relabelling with at most three common edges. Consequently, \[ 2n-10n^{2/3}-7<f(n,4)\leq 2n-4, \] and hence \[ f(n,4)=2n-o(n). \] Thus we resolve Erd\H{o}s's four-edge intersection problem asymptotically, confirming his proposed value up to a sublinear error term. For comparison, for all sufficiently large $n$, Fang and Hou's result guarantees at most four common edges for graphs with at most $2n-3$ edges, whereas reducing the edge bound by only $10n^{2/3}+4=o(n)$ already allows us to guarantee at most three common edges.
For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put \[ I_G(\sigma)=|E(G)\cap E(\sigma(G))|. \] Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In 1977 Erd\H{o}s...
For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erd\H{o}s and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)...
For graphs $H$ and $F$, let $ex(n,H,F)$ be the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We study this problem when $H$ is a clique and $F=T_t$which is a fixed tree on $t$ vertices. The Erd\H{o}s--S\'{o}s conjecture concerns the value of $ex(n,K_2, T_t)$. Gerbner and Palmer proposed a more genera...
Let $z(n)$ be the maximum number of edges in a tripartite $3$-uniform hypergraph with $n$ vertices in each part and no copy of $K_{2,2,2}^{(3)}$ (a ``box''). Erd\H{o}s (1964) proved that $z(n) = O(n^{11/4})$, whereas the best previous lower bound, due to Katz, Krop, and Maggioni (2002), was $\Omega(n^{8/3})$. For each...
A graph $G$ is $n$-existentially closed or $n$-e.c. if, for all subsets $S\subseteq V(G)$ with $|S|=n$ and for all partitions $S=A\sqcup B$, there exists a vertex in $V(G)\setminus S$ adjacent to all vertices in $A$ and no vertices in $B$. We study the minimum number of edges $m(v,n)$ of a $v$-vertex $n$-e.c. graph, an...
Isabel T. Byrne, John Byrne, S. Cioabă· 0 citations
For a graph $G$, let $e(G)$ denote the number of edges of $G$, and let $c(G)$ be the number of distinct cycles in $G$. Morrison, Roberts and Scott asked whether, for every fixed graph $H$ and all large $n$, some $n$-vertex $H$-free graph maximizes both $e(G)$ and $c(G)$. In this paper, we show that the answer is no for...
