[Theory] UC Theory Seminar: a reminder

[Alexander Razborov via Theory]

Maya Sankar
Stanford University
 
 
  
Tuesday, January 14, 2025 at 3:30pm
Kent Chemical Laboratory, Room 120
 
 
 
 Title: The Turán Density of 4-Uniform Tight Cycles
 
Abstract:  For any uniformity r and residue k modulo r, we give an exact characterization of the r-uniform hypergraphs that homomorphically avoid tight cycles of length k modulo r, in terms of colorings of (r-1)-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, as well as a result of Kamčev, Letzter, and Pokrovskiy in uniformity 3.  In fact, our characterization applies to a much larger class of families than those of the form C_k^(r)={r-uniform tight cycles of length k modulo r}.
 
We also outline a general strategy to prove that, if C is a family of tight-cycle-like hypergraphs (including but not limited to the families C_k^(r)) for which the above characterization applies, then all sufficiently long cycles in C will have the same Turán density. We demonstrate an application of this framework, proving that there exists an integer L_0 such that for every L>L_0 not divisible by 4, the tight cycle C^(4)_L has Turán density 1/2.
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