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MATH Seminar

Title: Off-diagonal Ramsey numbers for slowly growing hypergraphs
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 |eii1j=1ej|1 for each i{2,,t}. We prove that if k3 is fixed and F is any non k-partite slowly growing k-uniform hypergraph, then for n2, r(F,n)=Ω(nk(logn)2k2).\\ \\ 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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