11,216 research outputs found

    Tur\'an and Ramsey Properties of Subcube Intersection Graphs

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    The discrete cube {0,1}d\{0,1\}^d is a fundamental combinatorial structure. A subcube of {0,1}d\{0,1\}^d is a subset of 2k2^k of its points formed by fixing kk coordinates and allowing the remaining d−kd-k to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. 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+1r+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 kk which have non-empty intersection and no ll which are pairwise disjoint? These questions are naturally expressed as Tur\'an and Ramsey type questions in intersection graphs of subcubes where 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. Tur\'an and Ramsey type problems are at the heart of extremal combinatorics and so these problems are mathematically natural. However, a second motivation is a connection with some questions in social choice theory arising from a simple model of agreement in a society. Specifically, if we have to make a binary choice on each of nn separate issues then it is reasonable to assume that the set of choices which are acceptable to an individual will be represented by a subcube. Consequently, the pattern of intersections within a family of subcubes will have implications for the level of agreement within a society. We pose a number of questions and conjectures relating directly to the Tur\'an and Ramsey problems as well as raising some further directions for study of subcube intersection graphs.Comment: 18 page

    Integer colorings with forbidden rainbow sums

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    For a set of positive integers A⊆[n]A \subseteq [n], an rr-coloring of AA is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of [n][n] with the maximum number of rainbow sum-free rr-colorings. We show that for r=3r=3, the interval [n][n] is optimal, while for r≥8r\geq8, the set [⌊n/2⌋,n][\lfloor n/2 \rfloor, n] is optimal. We also prove a stability theorem for r≥4r\geq4. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

    Optimal transportation, topology and uniqueness

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    The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff [14]: give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure \gamma outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph / antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.Comment: 36 pages, 6 figure
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