Pantelis Eleftheriou (University of Leeds)
- Date
- Wednesday 19 October 2022, 4:00 PM
- Location
- EC Stoner seminar room 8.90
- Category
- Logic Seminar
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) "linear Zarankiewicz bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in $M$
- (2) $M$ does not define an infinite field.
We prove that the following are equivalent:
- (1)' "linear Zarankiewicz bounds" hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in $M$.
- (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.