Large cliques in graphs with forbidden semi-induced structures

TL;DR

This paper proves that graphs with many cliques and forbidden semi-induced structures necessarily contain large cliques, improving previous bounds and connecting to VC-dimension concepts.

Contribution

It establishes a stronger linear dependence on the density parameter for the existence of large cliques in graphs with forbidden semi-induced substructures.

Findings

Graphs with many cliques and forbidden structures contain large cliques.

Improved bound from exponential to linear dependence on density parameter.

Connection to VC-dimension in graph theory.

Abstract

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