# Aktuelles

## Einladung zum Vortrag zu „ordered Ramsey numbers“

[13.08.2020]

Einen Vortrag zum Thema „On ordered Ramsey numbers of tripartite 3-uniform hypergraphs“ hält Prof. Martin Balko, Ph.D., von der Karls-Universität in Prag am Donnerstag dem 13. August. Der Vortrag wird um 15 Uhr im Gebäude 3, Hybridraum H 006, der FernUniversität in Hagen stattfinden und über Zoom gestreamt, sodass Interessierte auch online teilnehmen können. Prof. Balko ist zurzeit Gast im Lehrgebiet Diskrete Mathematik und Optimierung (Prof. Dr. Winfried Hochstättler).

Abstract:

On ordered Ramsey numbers of tripartite 3-uniform hypergraphs

For an integer k >= 2, an ordered k-uniform hypergraph H is a k-uniform hypergraph together with a fixed linear ordering of its vertex set. The ordered Ramsey number R(H,G) of two ordered k-uniform hypergraphs H and G is the smallest positive integer N such that every red-blue coloring of the hyperedges of the ordered complete k-uniform hypergraph K^(k)_N on N vertices contains a blue copy of H or a red copy of G.

The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs. In particular, we prove that for all positive integers d,n and for every ordered 3-uniform hypergraph H on n vertices with maximum degree d and with interval chromatic number 3 there is an epsilon=epsilon(d)>0 such that R(H,H) <= 2^(O(n^(2-epsilon))). In fact, we prove this upper bound for the number R(G,K^(3)_3(n)), where G is an ordered 3-uniform hypergraph with n vertices and maximum degree d and K^(3)_3(n) is the ordered complete tripartite hypergraph with consecutive color classes of size n. We show that this bound is not far from the truth by proving R(H,K^(3)_3(n)) >= 2^(Omega(n\log(n))) for some fixed ordered 3-uniform hypergraph H.

Joint work with Máté Vizer.