28 research outputs found
Two-part set systems
The two part Sperner theorem of Katona and Kleitman states that if is an
-element set with partition , and \cF is a family of subsets
of such that no two sets A, B \in \cF satisfy (or ) and for some , 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 , . Along the way, we prove the following new result which may
be of independent interest: let \cF, \cG be families of subsets of an
-element set such that \cF and \cG are both intersecting and
cross-Sperner, meaning that if A \in \cF and B \in \cG, then and . Then |\cF| +|\cG| < 2^{n-1} and there are
exponentially many examples showing that this bound is tight
Helly-type problems
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
LIPIcs
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
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
We prove general topological Radon type theorems for sets in ,
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 -nets as
well as a -theorem.
More precisely: Let be either , smooth real -manifold, or
a finite -dimensional simplicial complex. Then if is a finite
family of sets in such that is at most for all and , then the Radon's number of is bounded in terms of
and . Here if ;
if is a smooth real -manifold and not a surface, if is
a surface and if is a -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 be a finite family of open sets in a surface
such that for every , is
either empty, or path-connected. Then the fractional Helly number of is at most three. This also settles a conjecture of Holmsen, Kim, and Lee
about an existence of a -theorem for open subsets of a surface.Comment: 11 pages, 2 figure