We show that there is an absolute constant c > 0 c>0 such that the following holds. For every n > 1 n > 1 , there is a 5-uniform hypergraph on at least 2 2 c n 1 / 4 2^{2^{cn^{1/4}}} vertices with independence number at most n n , where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for k k -uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.
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On Ramsey numbers of hedgehogs
Abstract The hedgehog H t is a 3-uniform hypergraph on vertices $1, \ldots ,t + \left({\matrix{t \cr 2}}\right)$ such that, for any pair ( i , j ) with 1 ≤ i < j ≤ t , there exists a unique vertex k > t such that { i , j , k } is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog H t is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r ( H t ) = O ( t 2 ln t ).
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- Award ID(s):
- 1855635
- PAR ID:
- 10178082
- Date Published:
- Journal Name:
- Combinatorics, Probability and Computing
- Volume:
- 29
- Issue:
- 1
- ISSN:
- 0963-5483
- Page Range / eLocation ID:
- 101 to 112
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/s0963548319000312
