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

Artem Chernikov; Chuyin Jiang

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arXiv · 2605.18123

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

Artem Chernikov·Chuyin Jiang

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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.

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Artem Chernikov, Chuyin Jiang. 2026-05-18. Fractional Helly property and combinatorics of forking in NTP$_2$ theories. https://arxiv.org/abs/2605.18123

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