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

Date and Time: Tuesday, April 16, 2024, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Simona Boyadzhiyska (University of Birmingham)

Ramsey goodness of loose paths

The Ramsey number of a pair of graphs (G,H), denoted by R(G,H), is the smallest integer n such that, for every red/blue-coloring of the edges of the complete graph K_n, there exists a red copy of G or a blue copy of H. In the 1980s, Burr showed that, if G is large and connected, then R(G,H) is bounded below by (v(G)-1)(χ(H)-1)+σ(H), where χ(H) is the chromatic number of H and σ(H) stands for the minimum size of a color class over all proper χ(H)-colorings of H. We say that G is H-good if R(G,H) is equal to this general lower bound. This notion was first studied systematically by Burr and Erdős and has received considerable attention from researchers since its introduction. Among other results, it was shown by Burr that, for any graph H, every sufficiently long path is H-good.

These concepts generalize in the natural way to k-graphs, and in this talk we will explore the notion of Ramsey goodness when G is an ℓ-path for some ℓ∈[k-1]. We will show that, while long loose paths are not always H-good, they are very close to being H-good for every k-graph H. As we will see, this is in stark contrast to the behavior of ℓ-paths for larger ℓ.

This is joint work with Allan Lo.

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