University of Birmingham > Talks@bham > Combinatorics and Probability Seminar > Distinct degrees and homogeneous sets

Distinct degrees and homogeneous sets

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If you have a question about this talk, please contact Eoin Long.

In this talk I will discuss some recent work examining the extremal relationship between two well-studied graph parameters: the order of the largest homogeneous set in a graph G and the maximal number of distinct degrees appearing in an induced subgraph of G, denoted respectively by hom (G) and f(G).

Our main theorem improves estimates due to Bukh and Sudakov and to Narayanan and Tomon and shows that if G is an n-vertex graph with hom (G) at least n1/2 then f(G) > ( n / hom (G) )1 – o(1). The bound here is sharp up to the o(1)-term, and asymptotically solves a conjecture of Narayanan and Tomon. In particular, this implies that max { hom (G), f(G) } > n1/2 -o(1) for any n-vertex graph G, which is also sharp.

The relationship between these parameters changes when hom (G) < n1/2. I hope to discuss the suspected relationship in this other region, along with supporting results. Joint work with Eoin Long.

This talk is part of the Combinatorics and Probability Seminar series.

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