P. Pach

activity

20242026

collaborators

9 papers

math.CO2026

On the Extended 1-2-3 Conjecture of Pilz

Philippa Holdridge, Péter Pál Pach

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

math.CO2026

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

János Nagy, Péter Pál 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

Péter Pál Pach, Csaba Sándor

Let $ρ_{k} (N)$ 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 pr…

math.NT2025

On the density of Kravitz sets

Vsevolod Lev, Máté Matolcsi, Péter Pál Pach +1

We show that for a subset $A$ of the cyclic group of prime order $p > 3$ , if the sumset $A + A - 2 A$ is not the whole group, then $∣ A ∣ \leq \frac{2}{7} p$ . Besides combinatorial arguments, we…

math.CO2025

Equidistribution of subset sums

Péter Pál 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

János Nagy, Péter Pál Pach

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

Ernie Croot, Vsevolod F. Lev, Péter Pál Pach

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.

math.CO2024

Product representation of perfect cubes

Zsigmond György Fleiner, Márk Hunor Juhász, Blanka Kövér +2

Let $F_{k, d} (n)$ be the maximal size of a set $A \subseteq [n]$ such that the equation \[a_1a_2\dots a_k=x^d, \; a_1<a_2<\ldots<a_k\] has no solution with $a_1,a_2,\ldots,a_k\in {…

math.CO2024

Optimal bounds on the polynomial Schur's theorem

Jaehoon Kim, Hong Liu, Péter Pál Pach

Liu, Pach and Sándor recently characterized all polynomials $p (z)$ such that the equation $x + y = p (z)$ is $2$ -Ramsey, that is, any $2$ -coloring of $N$ contains infinitely ma…