28 research outputs found

    Two-part set systems

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    The two part Sperner theorem of Katona and Kleitman states that if XX is an nn-element set with partition X1∪X2X_1 \cup X_2, and \cF is a family of subsets of XX such that no two sets A, B \in \cF satisfy A⊂BA \subset B (or B⊂AB \subset A) and A∩Xi=B∩XiA \cap X_i=B \cap X_i for some ii, then |\cF| \le {n \choose \lfloor n/2 \rfloor}. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfiy various combinations of these properties on one or both parts X1X_1, X2X_2. Along the way, we prove the following new result which may be of independent interest: let \cF, \cG be families of subsets of an nn-element set such that \cF and \cG are both intersecting and cross-Sperner, meaning that if A \in \cF and B \in \cG, then A⊄BA \not\subset B and B⊄AB \not\subset A. Then |\cF| +|\cG| < 2^{n-1} and there are exponentially many examples showing that this bound is tight

    Helly-type problems

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    In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals

    Generalized forbidden subposet problems

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    LIPIcs

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    The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2

    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

    Bounding Radon's number via Betti numbers

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    We prove general topological Radon type theorems for sets in Rd\mathbb R^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly and colorful Helly theorems, and consequently an existence of weak ε\varepsilon-nets as well as a (p,q)(p,q)-theorem. More precisely: Let XX be either Rd\mathbb R^d, smooth real dd-manifold, or a finite dd-dimensional simplicial complex. Then if F\mathcal F is a finite family of sets in XX such that β~i(⋂G;Z2)\widetilde\beta_i(\bigcap \mathcal G; \mathbb Z_2) is at most bb for all i=0,1,…,ki=0,1,\ldots, k and G⊆F\mathcal G\subseteq \mathcal F, then the Radon's number of F\mathcal F is bounded in terms of bb and XX. Here k=⌈d2⌉−1k=\left\lceil\frac{d}{2}\right\rceil-1 if X=RdX=\mathbb R^d; k=d−1k=d-1 if XX is a smooth real dd-manifold and not a surface, k=0k=0 if XX is a surface and k=dk=d if XX is a dd-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F\mathcal F be a finite family of open sets in a surface SS such that for every G⊆F\mathcal G\subseteq \mathcal F, ⋂G\bigcap \mathcal G is either empty, or path-connected. Then the fractional Helly number of F\mathcal F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)(p,q)-theorem for open subsets of a surface.Comment: 11 pages, 2 figure
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