9 research outputs found

    A Sperner theorem on unrelated chains of subsets

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    Incomparable Copies of a Poset in the Boolean Lattice

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    Let (Formula presented.) be the poset generated by the subsets of [n] with the inclusion relation and let (Formula presented.) be a finite poset. We want to embed (Formula presented.) into (Formula presented.) as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for all finite posets (Formula presented.) as (Formula presented.), where (Formula presented.) denotes the minimal size of the convex hull of a copy of (Formula presented.). We discuss both weak and strong (induced) embeddings

    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)

    Posets and extremal set systems

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