Abstract
Building on recent work of Mattheus and Verstraëte, we establish a general connection between Ramsey numbers of the form (Formula presented.) for (Formula presented.) a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an (Formula presented.) by (Formula presented.) (Formula presented.) -matrix that does not have any matrix from a fixed finite family (Formula presented.) derived from (Formula presented.) as a submatrix. As an application, we give new lower bounds for the Ramsey numbers (Formula presented.) and (Formula presented.), namely, (Formula presented.) and (Formula presented.). We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of (Formula presented.) for any fixed integer (Formula presented.).
| Original language | English |
|---|---|
| Pages (from-to) | 2014-2023 |
| Number of pages | 10 |
| Journal | Bulletin of the London Mathematical Society |
| Volume | 56 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - Jun 2024 |
Bibliographical note
Funding Information:The work of David Conlon was supported by NSF Grant DMS\u20102054452. Sam Mattheus was a Visiting Scholar at UCSD supported by a Fulbright Visiting Scholar Fellowship and a Fellowship of the Belgian American Foundation. The work of Dhruv Mubayi was supported by NSF Grants DMS\u20101763317, DMS\u20101952767, DMS\u20102153576, a Humboldt Research Award, and a Simons Fellowship. Jacques Verstra\u00EBte was supported by NSF Grant DMS\u20101952786.
Publisher Copyright:
© 2024 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
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