# Combinatorics Seminar - Arnau Padrol

16:15-18:00

Title: The convex dimension of hypergraphs and the hypersimplicial Van
Kampen-Flores Theorem

Abstract: I will present joint work with Leonardo Martínez-Sandoval on the
hypersimplicial generalization of the linear van Kampen-Flores theorem:
for each n, k and i we determine onto which dimensions can the
(n,k)-hypersimplex be linearly projected while preserving its
i-skeleton. This is motivated by the study of the convex dimensions of
hypergraphs. The convex dimension of a k-uniform hypergraph is the
smallest dimension d for which there is an injective mapping of its
vertices into R^d such that the set of k-barycenters of all hyperedges
is in convex position. Our results completely determine the convex
dimension of complete k-uniform hypergraphs. This settles an open
question by Halman, Onn and Rothblum, who solved the problem for
complete graphs. We also provide lower and upper bounds for the extremal
problem of estimating the maximal number of hyperedges of k-uniform
hypergraphs on n vertices with convex dimension d.