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Forbidden Hypermatrices Imply General Bounds on Induced Forbidden Subposet Problems

Published online by Cambridge University Press:  15 February 2017

ABHISHEK METHUKU  and
DÖMÖTÖR PÁLVÖLGYI
ABHISHEK METHUKU
Affiliation:
Department of Mathematics, Central European University, 1051 Budapest, Hungary (e-mail: [email protected])
DÖMÖTÖR PÁLVÖLGYI
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WA, UK (e-mail: [email protected])
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Abstract

We prove that for every poset P, there is a constant CP such that the size of any family of subsets of {1, 2, . . ., n} that does not contain P as an induced subposet is at most

$$C_P{\binom{n}{\lfloor\gfrac{n}{2}\rfloor}},$$
settling a conjecture of Katona, and Lu and Milans. We obtain this bound by establishing a connection to the theory of forbidden submatrices and then applying a higher-dimensional variant of the Marcus–Tardos theorem, proved by Klazar and Marcus. We also give a new proof of their result.

Keywords

Primary 05D05Secondary 05A05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 4 , July 2017 , pp. 593 - 602
Copyright
Copyright © Cambridge University Press 2017 

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