Large cliques in graphs with forbidden semi-induced structures
Nannan Chen, Yulai Ma, Fan Yang
TL;DR
The paper addresses finding large cliques in $n$-vertex graphs that contain many copies of $K_r$ while forbidding semi-induced substructures related to $K_r[2]$. It strengthens previous bounds by achieving a linear-in-$c$ dependence on the clique size under a bounded set of semi-induced forbidden configurations $\,\mathcal{K}_r^{[2]}\$, extending Holmsen's result. The authors employ VC-dimension arguments on the family of maximal cliques and a Sauer–Shelah type bound to control pattern complexity, combined with a probabilistic averaging contradiction to force a clique of size at least $\frac{c}{18r}n$ for large $n$. This work ties semi-induced forbidden structures to VC-dimension and opens questions about the minimal number of forbidden patterns needed for linear-in-$c$ bounds, suggesting broader applicability of the approach.
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 \(Ω(c^{2^{r-1}} n)\). 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 $Ω(cn)$. 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.