P. Pach

activity

20242026

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4 papers · 1 filter

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…