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

Date and Time: Tuesday, February 28, 2023, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Kalina Petrova

Size-Ramsey numbers of structurally sparse graphs

The size-Ramsey number of a graph H is the minimum number of edges of a host graph G such that any k-edge-colouring of G contains a monochromatic copy of H. Research has mainly focused on the size-Ramsey numbers of n-vertex graphs with constant maximum degree D. For example, graphs which also have constant treewidth are known to have linear size-Ramsey numbers. On the other extreme, the canonical examples of graphs of unbounded treewidth are the grid graphs, for which the best known bound has only very recently been improved from O(n^{3/2}) to O(n^{5/4}) by Conlon, Nenadov and Trujic. In this work, we prove a common generalization of these results by establishing new bounds on the size-Ramsey numbers in terms of treewidth (which may grow as a function of n). As a special case, this yields a bound of \tilde{O}(n^{3/2 - 1/(2D)}) for proper minor-closed classes of graphs. In particular, this bound applies to planar graphs, addressing a question of Wood. Our proof combines methods from structural graph theory and classic Ramsey-theorertic embedding techniques, taking advantage of the product structure exhibited by graphs with bounded treewidth. This is joint work with Nemanja Draganic, Marc Kaufmann, David Munha Correia, and Raphael Steiner.

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