Asymptotic stability of clique-saturating edges near Turán thresholds

Abstract

Let $f_{p+1}(n,m)$ denote the minimum number of $K_{p+1}$-saturating edges in an $n$-vertex $K_{p+1}$-free graph with $m$ edges. In this paper, we investigate the stability of this parameter near the Turán thresholds, with particular emphasis on the role of the minimum degree. We introduce the refined parameter $f_{p+1,p}(n,m,δ)$. It is defined for $K_{p+1}$-free graphs on $n$ vertices with $m$ edges, minimum degree $δ$, and containing at least a copy of $K_p$. For every fixed integer $p\ge3$, we establish a stability result when the number of edges is $\mathrm{ex}(n,K_p)+o(n^2)$, showing that \[ f_{p+1,p} (n,m,δ)=\left(\frac{(p-2)^2}{2(p-1)^3}+o(1)\right)n^2, \ \text{for }\ m=\mathrm{ex}(n,K_p)+o(n^2), \ δ\geq \left(\frac{p-2}{p-1}+o(1)\right)n. \] We further obtain a stability theorem in the regime where the number of edges is quadratically smaller than $\operatorname{ex}(n,K_{p+1})$. More precisely, for $0 \leq c< \frac{1}{2p(3p-1)}$, \[ f_{p+1,p} (n,m,δ)=\left(\frac{1}{2p}+o(1)\right)n^2, \ \text{for }\ m=\operatorname{ex}(n,K_{p+1})-(c+o(1))n^2, \ δ\geq \left(\frac{p-1}{p}-2c+o(1)\right)n. \]