2,665 research outputs found

    Q2Q_2-free families in the Boolean lattice

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    For a family F\mathcal{F} of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that F\mathcal{F} is P-free if it does not contain a subposet isomorphic to P. Let ex(n,P)ex(n, P) be the largest size of a P-free family of subsets of [n]. Let Q2Q_2 be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that 2No(N)ex(n,Q2)2.283261N+o(N),2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N), where N=(nn/2)N = \binom{n}{\lfloor n/2 \rfloor}. We also prove that the largest Q2Q_2-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.Comment: 18 pages, 2 figure

    Rainbow Ramsey problems for the Boolean lattice

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    We address the following rainbow Ramsey problem: For posets P,QP,Q what is the smallest number nn such that any coloring of the elements of the Boolean lattice BnB_n either admits a monochromatic copy of PP or a rainbow copy of QQ. We consider both weak and strong (non-induced and induced) versions of this problem. We also investigate related problems on (partial) kk-colorings of BnB_n that do not admit rainbow antichains of size kk

    An upper bound on the size of diamond-free families of sets

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    Let La(n,P)La(n,P) be the maximum size of a family of subsets of [n]={1,2,...,n}[n]=\{1,2,...,n\} not containing PP as a (weak) subposet. The diamond poset, denoted B2B_{2}, is defined on four elements x,y,z,wx,y,z,w with the relations x<y,zx<y,z and y,z<wy,z<w. La(n,P)La(n,P) has been studied for many posets; one of the major open problems is determining La(n,B2)La(n,B_{2}). Studying the average number of sets from a family of subsets of [n][n] on a maximal chain in the Boolean lattice 2[n]2^{[n]} has been a fruitful method. We use a partitioning of the maximal chains and introduce an induction method to show that La(n,B2)(2.20711+o(1))(nn2)La(n,B_{2})\leq(2.20711+o(1))\binom{n}{\left\lfloor \frac{n}{2}\right\rfloor }, improving on the earlier bound of (2.25+o(1))(nn2)(2.25+o(1))\binom{n}{\left\lfloor \frac{n}{2}\right\rfloor } by Kramer, Martin and Young.Comment: Accepted by JCTA. Writing is improved based on the suggestions of referee

    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, (nn2)\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, 12k1(nn2)\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 (nn2)\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 (k1)(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)
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