Distinct degrees in induced subgraphs

Abstract

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An N N -vertex graph is called C C -Ramsey if it has no homogeneous set of size C log ⁡ N C\log N . A theorem of Bukh and Sudakov, solving a conjecture of Erdős, Faudree, and Sós, shows that any C C -Ramsey N N -vertex graph contains an induced subgraph with Ω C ( N 1 / 2 ) \Omega _C(N^{1/2}) distinct degrees. We improve this to Ω C ( N 2 / 3 ) \Omega _C(N^{2/3}) , which is tight up to the constant factor. We also show that any N N -vertex graph with N > ( k − 1 ) ( n − 1 ) N > (k-1)(n-1) and n ≥ n 0 ( k ) = Ω ( k 9 ) n\geq n_0(k) = \Omega (k^9) either contains a homogeneous set of order n n or an induced subgraph with k k distinct degrees. The lower bound on N N here is sharp, as shown by an appropriate Turán graph, and confirms a conjecture of Narayanan and Tomon.