The Erdos-Mullin Five-Edge Intersection Problem

Abstract: For an $n$-vertex graph $G$ and a permutation $π$ of its vertex set, let $I_G(π)=|E(G)\cap E(πG)|$, and let $μ(G)=\min_π I_G(π)$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph $G$ satisfying $μ(G)\ge k$. Erdős recorded a construction of Mullin showing $f(n,5)\le 2n-2$ and asked whether equality holds for sufficiently large $n$. We prove that it does: $f(n,5)=2n-2$ for all sufficiently large $n$. Equivalently, every sufficiently large $n$-vertex graph with at most $2n-3$ edges admits a relabelling with at most four common edges. The proof combines a quantitative exclusion of almost-universal vertices, a finite high-degree core with low-degree buffer vertices, list packing, and a sparse permutation version of the Lovász local lemma.