MATH Seminar
Title: Off-diagonal Ramsey numbers for slowly growing hypergraphs |
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Seminar: Combinatorics |
Speaker: Jiaxi Nie, Postdoctoral Fellow of Georgia Tech |
Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
Date: 2024-10-31 at 4:00PM |
Venue: MSC W303 |
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Abstract: 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 {\em slowly growing} if there is an ordering e1,e2,…,et of its edges such that |ei∖⋃i−1j=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(nk(logn)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. |
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