Generalized Ramsey–Turán density for cliques
We study the generalized Ramsey–Turán function $\mathrm {RT}(n,K_s,K_t,o(n))$ , which is the maximum possible number of copies of $K_s$ in an n-vertex $K_t$ -free graph with independence number $o(n)$ . The case when $s=2$ was settled by Erdős, Sós, Bollobás, Hajnal,...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000295/type/journal_article |
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| author | Jun Gao Suyun Jiang Hong Liu Maya Sankar |
| author_facet | Jun Gao Suyun Jiang Hong Liu Maya Sankar |
| author_sort | Jun Gao |
| collection | DOAJ |
| description | We study the generalized Ramsey–Turán function
$\mathrm {RT}(n,K_s,K_t,o(n))$
, which is the maximum possible number of copies of
$K_s$
in an n-vertex
$K_t$
-free graph with independence number
$o(n)$
. The case when
$s=2$
was settled by Erdős, Sós, Bollobás, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all
$s\ge 3$
, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when t is much larger than s. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold. |
| format | Article |
| id | doaj-art-7ca5ad8257cd42919b5b44badf046ea5 |
| institution | OA Journals |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-7ca5ad8257cd42919b5b44badf046ea52025-08-20T02:18:38ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.29Generalized Ramsey–Turán density for cliquesJun Gao0https://orcid.org/0000-0002-4229-4508Suyun Jiang1https://orcid.org/0000-0002-6512-2848Hong Liu2https://orcid.org/0000-0002-5735-7321Maya Sankar3https://orcid.org/0009-0004-8763-431XExtremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, 34126, South Korea; E-mail: ,School of Artificial Intelligence, Jianghan University, Wuhan, 430056, China; E-mail:Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, 34126, South Korea; E-mail: ,Department of Mathematics, Stanford University, Stanford, CA 94305, USAWe study the generalized Ramsey–Turán function $\mathrm {RT}(n,K_s,K_t,o(n))$ , which is the maximum possible number of copies of $K_s$ in an n-vertex $K_t$ -free graph with independence number $o(n)$ . The case when $s=2$ was settled by Erdős, Sós, Bollobás, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all $s\ge 3$ , showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when t is much larger than s. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold.https://www.cambridge.org/core/product/identifier/S2050509425000295/type/journal_article05C3505D10 |
| spellingShingle | Jun Gao Suyun Jiang Hong Liu Maya Sankar Generalized Ramsey–Turán density for cliques Forum of Mathematics, Sigma 05C35 05D10 |
| title | Generalized Ramsey–Turán density for cliques |
| title_full | Generalized Ramsey–Turán density for cliques |
| title_fullStr | Generalized Ramsey–Turán density for cliques |
| title_full_unstemmed | Generalized Ramsey–Turán density for cliques |
| title_short | Generalized Ramsey–Turán density for cliques |
| title_sort | generalized ramsey turan density for cliques |
| topic | 05C35 05D10 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425000295/type/journal_article |
| work_keys_str_mv | AT jungao generalizedramseyturandensityforcliques AT suyunjiang generalizedramseyturandensityforcliques AT hongliu generalizedramseyturandensityforcliques AT mayasankar generalizedramseyturandensityforcliques |