On problems of Erd\H{o}s and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs

TL;DR

This paper uses flag algebra techniques to determine the minimum density of large cliques in graphs with specific forbidden subgraphs, resolving several cases of classical extremal graph theory problems.

Contribution

It provides exact asymptotic minimum densities for $s$-cliques in graphs with forbidden independent sets and cycles, and characterizes extremal graph structures.

Findings

Minimum 8-clique density in graphs without independent sets of size 3 is approximately 491411/268435456.

Minimum $s$-clique density in graphs with no independent set of size 3 and no induced 5-cycle is $2^{1-s}+o(1)$ for $s=4,5,6.

Characterization of extremal and near-extremal graphs for these density problems.

Abstract

Using flag algebras, we prove that the minimum density of $8$-cliques in a large graph without an independent set of size $3$ is $491411/268435456+o(1)$, thus resolving a new case of an old problem of Erd\H{o}s [Magyar Tud. Akad. Mat. Kutat\'o Int. K\"ozl. 7 (1962) 459-464]. Also, we establish some other results of this type; for example, we show that the minimum $s$-clique density in a large graph with no independent set of size 3 nor an induced 5-cycle is $2^{1-s}+o(1)$ when $s=4,5,6$. For each of these results, we also describe the structure of all extremal and almost extremal graphs of large order $n$. These results are applied to give an asymptotic solution to a number of cases of the problem of Baumann and Briggs [Electronic J Comb 32 (2025) P1.22] which asks for the minimum number of $s$-cliques in an $n$-vertex graph in which every $k$-set spans a $t$-clique.