KAIST Discrete Math Seminar

The limits of converging sequences of graphs are natural objects from extremal graph theory that are also representable as measure-theoretic objects (graphons) or as algebraic objects (flag algebra homomorphisms). I will give an introduction to this theory via a geometric relative: its adaptation from graphs to a combinatorial encoding of planar point sets. This is based on joint work with Alfredo Hubard, Rémi de Joannis de Verclos, Jean-Sébastien Sereni and Jan Volec (http://drops.dagstuhl.de/opus/volltexte/2015/5126/). The talk will not assume any specific knowledge.

In combinatorial and computational geometry, the complexity of a system of sets is often studied via its shatter function. I will introduce these functions, and discuss how their asymptotic growth rate is governed from a single of its values, in the spirit of the classical notion of “Vapnik-Chernonenkis dimension” of hypergraphs. In particular, I will describe a probabilistic construction that refutes a conjecture of Bondy and Hajnal. This is joint work with Boris Bukh (https://arxiv.org/abs/1701.06632). The talk will start from first principles.

Helly’s theorem, a classical result in discrete geometry, asserts that if n>d convex subsets of R^d have empty intersection, some d+1 of them must already have empty intersection. I will discuss some topological generalizations of Helly’s theorem, where convexity is replaced by connectivity assumptions on the nonempty intersections, that lead to non-embeddability results of Borsuk-Ulam type and to variations on Leray’s acyclic cover theorem (or the Nerve theorem).

The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that Kn embeds in a closed surface M if and only if (n − 3)(n − 4) ≤ 6b1(M), where b1(M) is the first Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1 embeds in R2k if and only if n ≤ 2k + 1.
I will discuss a conjecture of Kuhnel that generalizes both the Heawood inequality and the van Kampen-Flores theorem, and present some partial results toward this conjecture.

I’ll explain how to construct, given a set system, an inclusion-exclusion formula valid for that set system and with size sensitive to the size of the Venn diagram of the set system. The main ingredient of the proof are random simplicial complexes. The talk will start from first principle.
This is joint work with J. Matousek, M. Tancer, Z. Safernova and P. Patak from Charles university.