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

Date and Time: Tuesday, November 07, 2023, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Aurelio Sulser

Ramsey numbers for multiple copies of sparse graphs

For a graph H and an integer n, we let nH denote the disjoint union of n copies of H. In 1975, Burr, Erdös, and Spencer initiated the study of Ramsey numbers for nH, one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant c = c(H) such that r(nH) = (2|H| - α(H))n + c, provided n is sufficiently large. Subsequently, Burr gave an implicit way of computing c and noted that this long term behaviour occurs when n is triply exponential in |H|. Recently, Bucić and Sudakov revived the problem and established an essentially tight bound on n by showing r(nH) follows this behaviour already when the number of copies is just a single exponential. Together with Milos Trujić, I established significantly stronger bounds on n in case H is a sparse graph, most notably of bounded maximum degree. These are relatable to the current state of the art bounds on r(H) and (in a way) tight. Our methods rely on a beautiful classic proof of Graham, Rödl, and Rucinski, with the emphasis on developing an efficient absorbing method for bounded degree graphs.

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