arXiv subjects

Mervyn Tong

Publications and source records attributed to Mervyn Tong.

5 recordsLinked to original sources

arXiv2026

Combinatorics in 1-semi-equational theories

We give direct combinatorial proofs that 1-semi-equational theories satisfy two combinatorial properties that imply the non-interpretability of certain fields. Our main result is that Boolean combinations of 1-semi-equations have almost linear Zarankiewicz bounds, and hence no infinite field is interpretable in a 1-semi-equational theory; this answers a question of Chernikov--Mennen and Chernikov--Starchenko, who had respectively established almost linear Zarankiewicz bounds for Boolean combinations of $(2,1)$-semi-equations and Boolean combinations of weakly normal relations. (This was recently and independently established by Gou, Mirabi, Mittal, Tran, and Yang, using different techniques and producing different bounds.) We also show that $(k,1)$-semi-equations satisfy the $\delta$-strong Erd\H{o}s--Hajnal property with $\delta=1/6^{k-1}$; this had previously been established by Chernikov--Starchenko for an ineffective constant $\delta>0$.

math.LO↗

arXiv2026

Homogeneous hypergraph regularity lemmas via $k$-strong honest definitions

We prove that $(k+1)$-uniform hypergraphs definable in an NIP strongly $k$-distal structure satisfy a homogeneous regularity lemma -- they can be partitioned into a bounded number of simplicial complexes, most of which are homogeneous (meaning that the restriction of the hypergraph to the simplicial complex is either complete or empty). Furthermore, the parts of the partition can be chosen uniformly definably, and the size of the partition is polynomial in the reciprocal of the error parameter. This extends the homogeneous regularity lemma proven by Chernikov and Starchenko for hypergraphs definable in a distal structure. We prove this by introducing $k$-strong honest definitions and showing that an NIP structure is strongly $k$-distal if and only if every formula $\varphi(x_1, ..., x_k; y)$ has a $k$-strong honest definition. This extends the theory of strong honest definitions in distal structures to the higher-arity setting.

math.LO↗

arXiv2026

Higher-arity distality and forking triviality

Answering a question of Goode, we show that $k$-triviality collapses to (1-)triviality among simple theories. In particular, every stable theory with quantifier elimination in a relational language of bounded arity is trivial. We use our collapse result, along with other facts about $k$-triviality and $k$-total triviality, to generate examples of (strongly) $k$-distal theories. The collapse result immediately implies that no stable theory can be strictly $k$-distal for some $k\geq 3$, partially answering a question of Walker. Moreover, all known examples of non-distal (strongly) $k$-distal theories are $k$-ary, rendering (strong) $k$-distality moot as a $(k+1)$-ary dividing line; we give four classes of examples that are not $k$-ary. We also show that just as distality is not preserved under taking reducts, neither is (strong) $k$-distality.

math.LO↗

arXiv2024

Zarankiewicz bounds from distal regularity lemma

Since K\H{o}v\'ari, S\'os, and Tur\'an proved upper bounds for the Zarankiewicz problem in 1954, much work has been undertaken to improve these bounds, and some have done so by restricting to particular classes of graphs. In 2017, Fox, Pach, Sheffer, Suk, and Zahl proved better bounds for semialgebraic binary relations, and this work was extended by Do in the following year to arbitrary semialgebraic relations. In this paper, we show that Zarankiewicz bounds in the shape of Do's are enjoyed by all relations satisfying the distal regularity lemma, an improved version of the Szemer\'edi regularity lemma satisfied by relations definable in distal structures (a vast generalisation of o-minimal structures).

math.CO↗

arXiv2024

Distal expansions of Presburger arithmetic by a sparse predicate

We prove that the structure $(\mathbb{Z},<,+,R)$ is distal for all congruence-periodic sparse predicates $R\subseteq\mathbb{N}$. We do so by constructing strong honest definitions for representative formulas of the theory, providing a rare example of concrete distal decompositions.

math.LO↗