P. Pach

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20242026

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7 papers · 1 filter

math.CO2026

On the Extended 1-2-3 Conjecture of Pilz

Philippa Holdridge, PÃ©ter PÃ¡l Pach

We resolve (for all sufficiently large $n$ ) a conjecture of Pilz on the symmetric difference $A I^ (2 A) I^ \dots I^ (n A)$ for finite sets $A \subseteq N$ of positive integers. W…

math.CO2026

On a group ring identity related to the Alon-Jaeger-Tarsi conjecture

JÃ¡nos Nagy, PÃ©ter PÃ¡l Pach

In this note we formulate a conjecture about two group ring identities and prove that it would imply the Alon-Jaeger-Tarsi conjecture.

math.CO2026

Product representations of perfect powers

PÃ©ter PÃ¡l Pach, Csaba SÃ¡ndor

Let $I ¨ _{k} (N)$ denote the maximum size of a set $A \subseteq {1, 2, \dots, N}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$ -th power. In this short note, we p…

math.CO2025

Equidistribution of subset sums

PÃ©ter PÃ¡l Pach

We answer a question of Katona and Makar-Limanov, by showing that in an abelian group of order $2 h$ the $h$ -element subset sums are asymptotically (as $h \to \infty$ ) equidistribute…

math.CO2024

A sumset version of a conjecture of Pilz

Pilz's conjecture states that for any finite set $A = {a_{1}, a_{2}, \dots, a_{k}}$ of positive integers and positive integer $n$ in the union of the sets $\{a_1,2a_1,\dots,na_1\},\dots, \{…

math.CO2024

Past and future of the cap set problem

We survey the history of the capset problem in the context of related results on progression-free sets, discuss recent progress, and mention further directions to explore.