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Turán and Ramsey Properties of Subcube Intersection Graphs

Published online by Cambridge University Press:  03 October 2012

J. ROBERT JOHNSON  and
KLAS MARKSTRÖM
J. ROBERT JOHNSON
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK (e-mail: r.johnson@qmul.ac.uk)
KLAS MARKSTRÖM
Affiliation:
Department of Mathematics and Mathematical Statistics, Umeå University, Umeå 901 87, Sweden (e-mail: klas.markstrom@math.umu.se)
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Abstract

The discrete cube {0, 1}d is a fundamental combinatorial structure. A subcube of {0, 1}d is a subset of 2k of its points formed by fixing k coordinates and allowing the remaining d − k to vary freely. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no r + 1 of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no k which have non-empty intersection and no l which are pairwise disjoint?

These questions are naturally expressed using intersection graphs. The intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Let $\I(n,d)$ be the set of all n vertex graphs which can be represented as the intersection graphs of subcubes in {0, 1}d. With this notation our first question above asks for the largest number of edges in a Kr+1-free graph in $\I(n,d)$. As such it is a Turán-type problem. We answer this question asymptotically for some ranges of r and d. More precisely we show that if $(k+1)2^{\lfloor\frac{d}{k+1}\rfloor}<n\leq k2^{\lfloor\frac{d}{k}\rfloor}$ for some integer k ≥ 2 then the maximum edge density is $\bigl(1-\frac{1}{k}-o(1)\bigr)$ provided that n is not too close to the lower limit of the range.

The second question can be thought of as a Ramsey-type problem. The maximum such n can be defined in the same way as the usual Ramsey number but only considering graphs which are in $\I(n,d)$. We give bounds for this maximum n mainly concentrating on the case that l is fixed, and make some comparisons with the usual Ramsey number.

Keywords

Primary 05D05Secondary 05C3505C62
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 1 , January 2013 , pp. 55 - 70
Copyright
Copyright © Cambridge University Press 2012

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