28 research outputs found
Discrete Geometry
A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry
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
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
Exact Bounds for Some Hypergraph Saturation Problems
Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G
on vertex sets X,Y that satisfies the following condition; one can add the
edges between X and Y that do not belong to G one after the other so that
whenever a new edge is added, a new copy of K_{p,q} is created. The problem of
bounding W_n(p,q), and its natural hypergraph generalization, was introduced by
Balogh, Bollob\'as, Morris and Riordan. Their main result, specialized to
graphs, used algebraic methods to determine W_n(1,q).
Our main results in this paper give exact bounds for W_n(p,q), its hypergraph
analogue, as well as for a new variant of Bollob\'as's Two Families theorem. In
particular, we completely determine W_n(p,q), showing that if 1 <= p <= q <= n
then
W_n(p,q) = n^2 - (n-p+1)^2 + (q-p)^2.
Our proof applies a reduction to a multi-partite version of the Two Families
theorem obtained by Alon. While the reduction is combinatorial, the main idea
behind it is algebraic
Quantitative Fractional Helly and (p,q)-Theorems
We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main geometric result is a quantitative version of the Fractional Helly Theorem of Katchalski and Liu, the second one is a quantitative version of the (p,q)-Theorem of Alon and Kleitman
Bounding Radon Number via Betti Numbers
We prove general topological Radon-type theorems for sets in ?^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ?-nets as well as a (p,q)-theorem.
More precisely: Let X be either ?^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ?? coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ?^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-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 be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface