Geometric constructions for Ramsey-Turán theory
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866912540058976256 |
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| author | Liu, Hong Reiher, Christian Sharifzadeh, Maryam Staden, Katherine |
| author_facet | Liu, Hong Reiher, Christian Sharifzadeh, Maryam Staden, Katherine |
| contents | Combining two classical notions in extremal combinatorics, the study of Ramsey-Turán theory seeks to determine, for integers $m\le n$ and $p \leq q$, the number $\mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$.
Two major open problems in this area from the 80s ask: (1) whether the asymptotic extremal structure for the general case exhibits certain periodic behaviour, resembling that of the special case when $p=2$; (2) constructing analogues of Bollobás-Erdős graphs with densities other than $1/2$.
We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the $p=2$ case, and address the second problem by constructing Bollobás-Erdős-type graphs using high dimensional complex spheres with all rational densities. Some matching upper bounds are also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_10423 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Geometric constructions for Ramsey-Turán theory Liu, Hong Reiher, Christian Sharifzadeh, Maryam Staden, Katherine Combinatorics Combining two classical notions in extremal combinatorics, the study of Ramsey-Turán theory seeks to determine, for integers $m\le n$ and $p \leq q$, the number $\mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$. Two major open problems in this area from the 80s ask: (1) whether the asymptotic extremal structure for the general case exhibits certain periodic behaviour, resembling that of the special case when $p=2$; (2) constructing analogues of Bollobás-Erdős graphs with densities other than $1/2$. We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the $p=2$ case, and address the second problem by constructing Bollobás-Erdős-type graphs using high dimensional complex spheres with all rational densities. Some matching upper bounds are also provided. |
| title | Geometric constructions for Ramsey-Turán theory |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2103.10423 |