31 research outputs found

    Sperner's problem for G-independent families

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    Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edge-less graph, this problem is resolved by Sperner's Theorem. In this paper, we focus on the case where G is the path of length n-1, proving the size of a maximal antichain is of the same order as the size of a largest layer of Q(G).Comment: 26 page

    Kernels, Stable Matchings, and Scarf's Lemma

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    Extremal problems on counting combinatorial structures

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    The fast developing field of extremal combinatorics provides a diverse spectrum of powerful tools with many applications to economics, computer science, and optimization theory. In this thesis, we focus on counting and coloring problems in this field. The complete balanced bipartite graph on nn vertices has \floor{n^2/4} edges. Since all of its subgraphs are triangle-free, the number of (labeled) triangle-free graphs on nn vertices is at least 2^{\floor{n^2/4}}. This was shown to be the correct order of magnitude in a celebrated paper Erd\H{o}s, Kleitman, and Rothschild from 1976, where the authors furthermore proved that almost all triangle-free graphs are bipartite. In Chapters 2 and 3 we study analogous problems for triangle-free graphs that are maximal with respect to inclusion. In Chapter 2, we solve the following problem of Paul Erd\H{o}s: Determine or estimate the number of maximal triangle-free graphs on nn vertices. We show that the number of maximal triangle-free graphs is at most 2n2/8+o(n2)2^{n^2/8+o(n^2)}, which matches the previously known lower bound. Our proof uses among other tools the Ruzsa-Szemer\'{e}di Triangle Removal Lemma and recent results on characterizing of the structure of independent sets in hypergraphs. This is a joint work with J\'{o}zsef Balogh. In Chapter 3, we investigate the structure of maximal triangle-free graphs. We prove that almost all maximal triangle-free graphs admit a vertex partition (X,Y)(X, Y) such that G[X]G[X] is a perfect matching and YY is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di Removal Lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on the characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint P3P_3's, which is of independent interest. This is a joint work with J\'{o}zsef Balogh, Hong Liu, and Maryam Sharifzadeh. In Chapte 4, we seek families in posets with the smallest number of comparable pairs. Given a poset PP, a family \F\subseteq P is \emph{centered} if it is obtained by `taking sets as close to the middle layer as possible'. A poset PP is said to have the \emph{centeredness property} if for any MM, among all families of size MM in PP, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice {0,1}n\{0,1\}^n has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset {0,1,
,k}n\{0,1,\ldots,k\}^n also has the centeredness property, provided nn is sufficiently large compared to kk. We show that this conjecture is false for all k≄2k\geq 2 and investigate the range of MM for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of Fqn\mathbb{F}_q^n has the centeredness property. Several open problems are also given. This is a joint result with J\'{o}zsef Balogh and Adam Zsolt Wagner. In Chapter 5, we consider a graph coloring problem. Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some kk such that all kk-th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant cc such that for any kk there is a family of graphs GG with χ(Gk)\chi(G^k) unbounded and χℓ(Gk)≄cχ(Gk)logâĄÏ‡(Gk)\chi_{\ell}(G^k)\geq c \chi(G^k) \log \chi(G^k). We also provide an upper bound, χℓ(Gk)1\chi_{\ell}(G^k)1. This is a joint work with Nicholas Kosar, Benjamin Reiniger, and Elyse Yeager

    Maximum number of colors in hypertrees of bounded degree

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    The upper chromatic number (Formula presented.) of a hypergraph (Formula presented.) is the maximum number of colors that can occur in a vertex coloring (Formula presented.) such that no edge (Formula presented.) is completely multicolored. A hypertree (also called arboreal hypergraph) is a hypergraph whose edges induce subtrees on a fixed tree graph. It has been shown that on hypertrees it is algorithmically hard not only to determine exactly but also to approximate the value of (Formula presented.), unless (Formula presented.). In sharp contrast to this, here we prove that if the input is restricted to hypertrees (Formula presented.) of bounded maximum vertex degree, then (Formula presented.) can be determined in linear time if an underlying tree is also given in the input. Consequently, (Formula presented.) on hypertrees is fixed parameter tractable in terms of maximum degree. © 2014 Springer-Verlag Berlin Heidelberg

    Finding lower bounds on the complexity of secret sharing schemes by linear programming

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    Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing schemePeer ReviewedPostprint (author's final draft

    Generalizations of Sperner\u27s Theorem: Packing Posets, Families Forbidding Posets, and Supersaturation

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    Sperner\u27s Theorem is a well known theorem in extremal set theory that gives the size of the largest antichain in the poset that is the Boolean lattice. This is equivalent to finding the largest family of subsets of an nn-set, [n]:={1,2,
,n}[n]:=\{1,2,\dots,n\}, such that the family is constructed from pairwise unrelated copies of the single element poset. For a poset PP, we are interested in maximizing the size of a family F\mathcal{F} of subsets of [n][n], where each maximally connected component of F\mathcal{F} is a copy of PP, and finding the extreme configurations that achieve this value. For instance, Sperner showed that when PP is one element, (n⌊n2⌋)\dbinom{n}{\lfloor \frac{n}{2}\rfloor} is the maximum number of copies of PP and that this is only achieved by taking subsets of a middle size. Griggs, Stahl, and Trotter have shown that when PP is a chain on kk elements, 12k−1(n⌊n2⌋)\dfrac{1}{2^{k-1}}\dbinom{n}{\lfloor \frac{n}{2}\rfloor} is asymptotically the maximum number of copies of PP. We find the extreme families for a packing of chains, answering a conjecture of Griggs, Stahl, and Trotter, as well as finding the extreme packings of certain other posets. For the general poset PP, we prove that the maximum number of unrelated copies of PP is asymptotic to a constant times (n⌊n2⌋)\dbinom{n}{\lfloor \frac{n}{2}\rfloor}. Moreover, the constant has the form 1c(P)\dfrac{1}{c(P)}, where c(P)c(P) is the size of the smallest convex closure over all embeddings of PP into the Boolean lattice. Sperner\u27s Theorem has been generalized by looking for La⁡(n,P)\operatorname{La}(n,P), the size of a largest family of subsets of an nn-set that does not contain a general poset PP in the family. We look at this generalization, exploring different techniques for finding an upper bound on La⁡(n,P)\operatorname{La}(n,P), where PP is the diamond. We also find all the families that achieve La⁡(n,{V,Λ})\operatorname{La}(n,\{\mathcal{V},\Lambda\}), the size of the largest family of subsets that do not contain either of the posets V\mathcal{V} or Λ\Lambda. We also consider another generalization of Sperner\u27s theorem, supersaturation, where we find how many copies of PP are in a family of a fixed size larger than La⁡(n,P)\operatorname{La}(n,P). We seek families of subsets of an nn-set of given size that contain the fewest kk-chains. Erd\H{o}s showed that a largest kk-chain-free family in the Boolean lattice is formed by taking all subsets of the (k−1)(k-1) middle sizes. Our result implies that by taking this family together with xx subsets of the kk-th middle size, we obtain a family with the minimum number of kk-chains, over all families of this size. We prove our result using the symmetric chain decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951)

    From Points to Potlucks: An Exploration of Fixed Point Theorems with Applications to Game Theory Models of Successful Integration Practices

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    Potlucks have many names: shared community dinners, faith suppers, “bring-a-dish” dinners, etc. They represent the desire to share food with other people and make new friends, sometimes learning about other cultures in the process. Not only does one have to decide what dish to bring, but one must also decide how large of a dish, if there will be a theme, and what course it will fit. For instance, if everyone brings side dishes, there will not be enough food for everyone, and if someone brings food that most of the group cannot eat, then feelings will be hurt on all sides. And in a way, having a potluck is similar to creating integration policies. Successful integration policies are fair to all people and take a “two-street” approach, while simultaneously being a collaborative affair. This paper will first explore fixed point theory, including the Kakutani Fixed Point Theorem and Brouwer Fixed Point Theorem; fixed point theorems are a significant field of mathematics and have many well-known applications. One of these applications is game theory, which is the study of how rational actors make decisions in everyday situations. Building upon the mathematical aspects of the first few chapters and the basics of game theory, this paper aims to build its own game theory model called the “Potluck Metaphor” that will model several methods of integration in the European Union; context for the model will be provided by critiquing three primary integration models and a brief literature review of the related field. Starting off with a simple game theory model for a dinner party, this paper will then slowly expand these models to show their applicability to European integration policy on an organizational level and on a member-specific level

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