English

Large cliques in graphs with forbidden semi-induced structures

Combinatorics 2025-11-18 v1

Abstract

In 2022, Holmsen showed that any graph with at least c(nr) c \binom{n}{r} rr-cliques but no induced complete rr-partite graph K2,,2K_{2,\ldots, 2} must contain a clique of order Ω(c2r1n)\Omega(c^{2^{r-1}} n). In this paper, we study graphs forbidding semi-induced substructures and show that every nn-vertex graph GG containing at least c(nr)c\binom{n}{r} copies of KrK_r (for some constant c>0c>0) and forbidding semi-induced substructures, related to K2,,2K_{2,\ldots, 2}, must contain a clique of order Ω(cn)\Omega(cn). Our result strengthens Holmsen's bound by improving the dependence on cc from c2r1c^{2^{r-1}} to linear in cc with bounded number of forbidden structures. Furthermore, our approach is naturally linked to the notion of VC-dimension.

Keywords

Cite

@article{arxiv.2511.13073,
  title  = {Large cliques in graphs with forbidden semi-induced structures},
  author = {Nannan Chen and Yulai Ma and Fan Yang},
  journal= {arXiv preprint arXiv:2511.13073},
  year   = {2025}
}
R2 v1 2026-07-01T07:40:38.802Z