Fractional Helly property and combinatorics of forking in NTP$_2$ theories

TL;DR

This paper explores FHP theories, a new subclass of low NTP$_2$ theories characterized by the Fractional Helly Property, providing new examples and analyzing forking and $f$-generics in these theories.

Contribution

It introduces FHP theories as a generalization of NIP, offers new examples including ultraproducts of finite fields and p-adics, and studies forking and $f$-generics within this framework.

Findings

FHP theories form a new subclass of low NTP$_2$ theories.

Provided new examples such as ultraproducts of finite fields and p-adics.

Established partial results on forking and $f$-generics in FHP theories.

Abstract

We investigate the class of FHP theories, i.e. theories of structures in which all definable families of sets satisfy the Fractional Helly Property (and its variants) from combinatorics. FHP theories generalize NIP and form a new subclass of low NTP$_2$ theories. We give many new examples (including ultraproducts of finite fields and of the $p$-adics) and establish some results about forking and $f$-generics for amenable groups definable in FHP theories. We make several conjectures about finitary combinatorial properties of forking in NTP$_2$ theories and establish some partial results, as well as investigate related two-cardinal type counting functions addressing a question of Adler.