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On problems of Erdős and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs

Levente Bodnár, Oleg Pikhurko

TL;DR

This work advances extremal graph theory by solving new cases of minimizing the density of $s$-cliques in large graphs under forbidden induced subgraphs, with a focus on graphs lacking an independent set of size $3$ and connections to the Baumann–Briggs problem. Leveraging Razborov's flag-algebra method, it computes exact asymptotic minimum densities such as $ER_{8}(n,\overline{K}_3)=\frac{491411}{268435456}+o(1)$ and shows that extremal structures are uniform expansions of specific Ramsey-type graphs like $R_{3,3,3}$, together with stability results. It further derives asymptotic densities for several Baumann–Briggs configurations by reducing to Erdős-type problems via the framework of $\mathcal{B}_{k,t}'$, yielding explicit values for $er_{s}(\mathcal{B}_{k,t})$ in multiple regimes and exact descriptions of large extremal graphs. Overall, the paper unifies Erdős’s and Baumann–Briggs’s questions through a robust flag-algebra approach and yields both exact densities and detailed structural characterizations of extremal graphs in a broad parameter range.

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 Erdős [Magyar Tud. Akad. Mat. Kutató Int. Közl. 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.

On problems of Erdős and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs

TL;DR

This work advances extremal graph theory by solving new cases of minimizing the density of -cliques in large graphs under forbidden induced subgraphs, with a focus on graphs lacking an independent set of size and connections to the Baumann–Briggs problem. Leveraging Razborov's flag-algebra method, it computes exact asymptotic minimum densities such as and shows that extremal structures are uniform expansions of specific Ramsey-type graphs like , together with stability results. It further derives asymptotic densities for several Baumann–Briggs configurations by reducing to Erdős-type problems via the framework of , yielding explicit values for in multiple regimes and exact descriptions of large extremal graphs. Overall, the paper unifies Erdős’s and Baumann–Briggs’s questions through a robust flag-algebra approach and yields both exact densities and detailed structural characterizations of extremal graphs in a broad parameter range.

Abstract

Using flag algebras, we prove that the minimum density of -cliques in a large graph without an independent set of size is , thus resolving a new case of an old problem of Erdős [Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962) 459-464]. Also, we establish some other results of this type; for example, we show that the minimum -clique density in a large graph with no independent set of size 3 nor an induced 5-cycle is when . For each of these results, we also describe the structure of all extremal and almost extremal graphs of large order . 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 -cliques in an -vertex graph in which every -set spans a -clique.
Paper Structure (4 sections, 17 theorems, 13 equations, 1 table)

This paper contains 4 sections, 17 theorems, 13 equations, 1 table.

Table of Contents

  1. Introduction
  2. Consequences for the Baumann--Briggs problem
  3. Proofs
  4. Concluding remarks

Key Result

Theorem 1

Let $(s,\ell)$ be one of the pairs of integers listed below and let $H$ be the following graph: Then every almost $\mathrm{ER}_{s}(n,\overline{K}_\ell)$-extremal graph is $o(n^2)$-close in the edit distance to a uniform expansion of $H$. Moreover, for all sufficiently large $n$, every $\mathrm{ER}_{s}(n,\overline{K}_\ell)$-extremal graph contains an expansion of $H$ as a spanning subgraph.

Theorems & Definitions (18)

  • Theorem 1: DasHuangMaNavesSudakov13Goodman59Lorden62ParczykPokuttaSpiegelSzabo23PikhurkoVaughan13
  • Theorem 2
  • Proposition 3
  • Lemma 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • Lemma 9
  • Corollary 10
  • ...and 8 more