Ramsey and Turán numbers of sparse hypergraphs
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929195714609152 |
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| author | Fox, Jacob Sankar, Maya Simkin, Michael Tidor, Jonathan Zhou, Yunkun |
| author_facet | Fox, Jacob Sankar, Maya Simkin, Michael Tidor, Jonathan Zhou, Yunkun |
| contents | Degeneracy plays an important role in understanding Turán- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a $k$-uniform hypergraph as the degeneracy of its $1$-skeleton (i.e., the graph formed by replacing every $k$-edge by a $k$-clique). We prove that skeletal degeneracy controls hypergraph Turán and Ramsey numbers in a similar manner to (graphical) degeneracy.
Specifically, we show that $k$-uniform hypergraphs with bounded skeletal degeneracy have linear Ramsey number. This is the hypergraph analogue of the Burr-Erdős conjecture (proved by Lee). In addition, we give upper and lower bounds of the same shape for the Turán number of a $k$-uniform $k$-partite hypergraph in terms of its skeletal degeneracy. The proofs of both results use the technique of dependent random choice. In addition, the proof of our Ramsey result uses the `random greedy process' introduced by Lee in his resolution of the Burr-Erdős conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00359 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ramsey and Turán numbers of sparse hypergraphs Fox, Jacob Sankar, Maya Simkin, Michael Tidor, Jonathan Zhou, Yunkun Combinatorics Degeneracy plays an important role in understanding Turán- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a $k$-uniform hypergraph as the degeneracy of its $1$-skeleton (i.e., the graph formed by replacing every $k$-edge by a $k$-clique). We prove that skeletal degeneracy controls hypergraph Turán and Ramsey numbers in a similar manner to (graphical) degeneracy. Specifically, we show that $k$-uniform hypergraphs with bounded skeletal degeneracy have linear Ramsey number. This is the hypergraph analogue of the Burr-Erdős conjecture (proved by Lee). In addition, we give upper and lower bounds of the same shape for the Turán number of a $k$-uniform $k$-partite hypergraph in terms of its skeletal degeneracy. The proofs of both results use the technique of dependent random choice. In addition, the proof of our Ramsey result uses the `random greedy process' introduced by Lee in his resolution of the Burr-Erdős conjecture. |
| title | Ramsey and Turán numbers of sparse hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.00359 |