Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, May 14, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Matija Bucic

Covering random graphs by monochromatic trees and Helly-type results for hypergraphs

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given r-edge-coloured graph G? These problems were introduced in the 1960's and were intensively studied by various researchers over the last 50 years. We consider the question of covering random graphs using monochromatic trees, introduced by Bal and DeBiasio. Surprisingly, it is closely connected to the following, independently interesting, Helly-type problem about vertex covers of hypergraphs. Roughly speaking, the question is how large a cover of a hypergraph H can be if any subgraph of H with few edges has a small cover. We prove good bounds for the hypergraph problem and use them to estimate quite accurately the number of monochromatic trees needed to cover a random graph. Our results provide some very surprising answers to several questions in the area asked by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo and Girao, Letzter and Sahasrabudhe. This is joint work with D. Korandi and B. Sudakov.

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