Polynomial to exponential transition in Ramsey theory

Hypergraph Ramsey numbers of cliques versus stars

“…The intermediate growth rate of $$$ r\left({K}_4^{(3)},{S}_n^{(3)}\right) $$$ is in striking contrast with the “polynomial‐to‐exponential transition” conjecture of Erdős and Hajnal [10], whose exact statement is technical but roughly states that all off‐diagonal hypergraph Ramsey numbers against cliques are either at most polynomial or at least exponential. This conjecture was proved to be true infinitely often when $$$ k=3 $$$ by Conlon, Fox and Sudakov [5] and was then settled in the affirmative for all $$$ k\ge 4 $$$ by Mubayi and Razborov [14]. As a corollary of Theorem 1.2, we see that no such transition can occur for hypergraph Ramsey numbers against stars.…”

Independent sets in hypergraphs with a forbidden link

“…Erdős and Hajnal further conjectured a specific value for h (k) 1 (s), the polynomial to exponential threshold, and Erdős later (see [5]) offered $500 to resolve this conjecture. This conjecture was solved for k = 3 and infinitely many s by Conlon, Fox, and Sudakov [7] and recently solved for k 4 by Mubayi and Razborov [19]. Mubayi and Suk [21] recently proved the general conjecture in the case s = k + 1.…”

C5 ${C}_{5}$ is almost a fractalizer

“…The notion of fractalizer can be extended to other structures. Mubayi and Razborov [24] showed that every tournament on $$k\ge 4$$ vertices whose edges are colored by $$\left(\genfrac{}{}{0.0pt}{}{k}{2}\right)$$ distinct colors is a fractalizer in the (F4) sense. They used this to determine the precise number where a certain Ramsey problem transitions from polynomial to exponential growth, settling a conjecture of Erdős and Hajnal [13] for all $$k\ge 4$$.…”