On Off-Diagonal Hypergraph Ramsey Numbers
Creators
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1.
California Institute of Technology
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2.
Stanford University
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3.
Rutgers, The State University of New Jersey
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4.
Georgia Institute of Technology
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5.
University of Illinois at Chicago
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6.
University of California, San Diego
Abstract
A fundamental problem in Ramsey theory is to determine the growth rate in terms of n of the Ramsey number r(H,Kn(3)) of a fixed 3-uniform hypergraph H versus the complete 3-uniform hypergraph with n vertices. We study this problem, proving two main results. First, we show that for a broad class of H, including links of odd cycles and tight cycles of length not divisible by three, r(H,Kn(3))≥2ΩH(n logn). This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph H for which r(H,Kn(3)) is superpolynomial in n. This provides the first example of a separation between r(H,Kn(3)) and r(H,Kn,n,n(3)), since the latter is known to be polynomial in n when H is linear.
Copyright and License
© The Author(s) 2025. Published by Oxford University Press. All rights reserved.
This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/pages/standard-publication-reuse-rights)
Acknowledgement
This research was initiated during a visit to the American Institute of Mathematics under their SQuaREs program.
Communicated by Benny Sudakov.
Funding
D.C. was supported by NSF Awards DMS-2054452 and DMS-2348859.
J.F. was supported by NSF Award DMS-2154129.
X.H. was supported by NSF Award DMS-2103154.
D.M. was partially supported by NSF Awards DMS-1763317, DMS-1952767, and DMS-2153576, by a Humboldt Research Award, and by a Simons Fellowship.
A.S. was supported by an NSF CAREER Award and by NSF Awards DMS-1952786 and DMS-2246847.
J.V. was supported by NSF Award DMS-1800332.
Additional details
Related works
- Is new version of
- Discussion Paper: arXiv:2404.02021 (arXiv)
Funding
- National Science Foundation
- DMS-2054452
- National Science Foundation
- DMS-2348859
- National Science Foundation
- DMS-2154129
- National Science Foundation
- DMS-2103154
- National Science Foundation
- DMS-1763317
- National Science Foundation
- DMS-1952767
- National Science Foundation
- DMS-2153576
- Alexander von Humboldt Foundation
- Simons Foundation
- National Science Foundation
- DMS-1952786
- National Science Foundation
- DMS-2246847
- National Science Foundation
- DMS-1800332
Dates
- Accepted
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2025-04-24
- Available
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2025-05-23Published