Graph and Hypergraph Counterexamples to Mubayi's supersaturation Conjecture Beyond $n^{k-1}$

Abstract

For a fixed $k$-graph $F$, let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex $k$-graph with $\text{ex}(n,F)+q$ edges. Let $c(n,F)$ be the minimum number created by adding one edge to an extremal $F$-free $k$-graph. We construct stable counterexamples to Mubayi's supersaturation conjecture for every $k\ge 2$. The ratio $h_F(n,q)/(qc(n,F))$ tends to zero over ranges extending beyond $n^{k-1}$. For graphs, we give examples of every chromatic number at least three. In particular, there is a graph $F$ for which this ratio tends to zero for $1\le q=o(n^{4/3})$. For $k\ge 3$, we use connected linear expansions of $G=(M_a\cup K_{1,b}\cup I_h)\vee K_{m_1,\dots ,m_{k-1}}$. Under explicit parameter conditions, the ratio tends to zero for $1\le q=o(n^{k-1+(b-1)/(a+b-1)})$ when $b\ge k+1$. We prove the exact Turán theorems for both constructions and determine their unique extremal graphs and hypergraphs. Keywords: supersaturation; extremal graph; linear hypergraph; graph expansion; stability.