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Pantelis Eleftheriou (University of Leeds)

Category
Logic Seminar
Date
Date
Wednesday 19 October 2022, 4:00 PM
Location
EC Stoner seminar room 8.90

An unbounded version of Zarankiewicz's problem

Note room change: EC Stoner seminar room 8.90

Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let $M$ be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:

  1. (1) "linear Zarankiewicz bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in $M$
  2. (2) $M$ does not define an infinite field.

We prove that the following are equivalent:

  1. (1)' "linear Zarankiewicz bounds" hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in $M$.
  2. (2)' $M$ does not define a full field (that is, one whose domain is the whole universe of $M$).

This is joint work (in progress) with Aris Papadopoulos.