On problems of Erdős and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs
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arXiv
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| Format: | Preprint |
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2026
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| author | Bodnár, Levente Pikhurko, Oleg |
| author_facet | Bodnár, Levente Pikhurko, Oleg |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17412 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On problems of Erdős and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs Bodnár, Levente Pikhurko, Oleg Combinatorics 05C35 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. |
| title | On problems of Erdős and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2602.17412 |