2024/09/02 by Sam Mattheus, Dhruv Mubayi, Mattheus, Sam +5 · 1 citation
Computer Science · Mathematics · #05D10 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2409.01442
openalex publication_date 2024/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a k-uniform hypergraph F and a positive integer n, the Ramsey number r(F,n) denotes the minimum N such that every N-vertex F-free k-uniform hypergraph contains an independent set of n vertices. A hypergraph is slowly growing if there is an ordering e1,e2,…,et of its edges such that |ei ∖ \bigcupj = 1i - 1ej| ≤ 1 for each i ∈ \2, …, t\. We prove that if k ≥ 3 is fixed and F is any non k-partite slowly growing k-uniform hypergraph, then for n≥2, r(F,n) = Ω(\fracnk(log n)2k - 2). In particular, we deduce that the off-diagonal Ramsey number r(F5,n) is of order n3/polylog(n), where F5 is the triple system \123, 124, 345\. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers.