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THE ERDŐS–SZEKERES PROBLEM AND AN INDUCED RAMSEY QUESTION

Part of: Extremal combinatorics

Published online by Cambridge University Press:  12 April 2019

Dhruv Mubayi  and
Andrew Suk
Dhruv Mubayi
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607, U.S.A. email mubayi@uic.edu
Andrew Suk
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, U.S.A. email asuk@ucsd.edu
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Abstract

Motivated by the Erdős–Szekeres convex polytope conjecture in $\mathbb{R}^{d}$, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers $n>k\geqslant 5$, what is the minimum integer $g_{k}(n)$ such that any$k$-uniform hypergraph on $g_{k}(n)$ vertices with the property that any set of $k+1$ vertices induces 0, 2, or 4 edges, contains an independent set of size $n$. Our main result shows that $g_{k}(n)>2^{cn^{k-4}}$, where $c=c(k)$.

MSC classification

Primary: 05D10: Ramsey theory

Information

Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 702 - 707
Copyright
Copyright © University College London 2019 

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Footnotes

The first author’s research was partially supported by NSF grant DMS-1763317. The second author was supported by an NSF CAREER award and an Alfred Sloan Fellowship.

References

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