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On Ramsey numbers of hedgehogs

Part of: Graph theory

Published online by Cambridge University Press:  18 October 2019

Jacob Fox  and
Ray Li
Jacob Fox
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
Ray Li*
Affiliation:
Department of Computer Science, Stanford University, Stanford, CA 94305, USA
*
*Corresponding author. Email: rayyli@cs.stanford.edu
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Abstract

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The hedgehog Ht 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 Ht is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(Ht) = O(t2 ln t).

MSC classification

Primary: 05C55: Generalized Ramsey theory
Secondary: 05C65: Hypergraphs

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 1 , January 2020 , pp. 101 - 112
Copyright
© Cambridge University Press 2019 

Footnotes

†

Research supported by a Packard Fellowship, and by NSF Career Award DMS-1352121.

‡

Research supported by the National Science Foundation Graduate Research Fellowship Program under grant DGE-1656518.

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