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≥3, we establish a stability result when the number of edges is ex(n,K_p)+o(n²), showing that f_p+1,p (n,m,δ)=(((p−2)²)/(2(p−1)³)+o(1))n², for m=ex(n,K_p)+o(n²), δ≥ ((p−2)/(p−1)+o(1))n. We further obtain a stability theorem in the regime where the number of edges is quadratically smaller than ex(n,K_p+1). More precisely, for 0 ≤ c< 1/(2p(3p−1)), f_p+1,p (n,m,δ)=(1/2p+o(1))n², for m=ex(n,K_p+1)−(c+o(1))n², δ≥ ((p−1)/p−2c+o(1))n.

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BibTeX RIS

BibTeX

@misc{chen2026asymptotic,
  title = {{Asymptotic stability of clique-saturating edges near Turán thresholds}},
  author = {Xianglei Chen and Changxiang He},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40696},
  eprint = {2610.40696},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40696},
}

RIS

TY  - UNPB
TI  - Asymptotic stability of clique-saturating edges near Turán thresholds
AU  - Xianglei Chen
AU  - Changxiang He
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.40696
UR  - https://arxiv.org/abs/2610.40696
AB  - 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≥3, we establish a stability result when the number of edges is ex(n,K_p)+o(n^2), showing that f_p+1,p (n,m,δ)=(((p−2)^2)/(2(p−1)^3)+o(1))n^2, for m=ex(n,K_p)+o(n^2), δ≥ ((p−2)/(p−1)+o(1))n. We further obtain a stability theorem in the regime where the number of edges is quadratically smaller than ex(n,K_p+1). More precisely, for 0 ≤ c< 1/(2p(3p−1)), f_p+1,p (n,m,δ)=(1/2p+o(1))n^2, for m=ex(n,K_p+1)−(c+o(1))n^2, δ≥ ((p−1)/p−2c+o(1))n.
ER  -

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