P'eter P'al Pach

activity

20112022

n

-smooth numbers with

n

colors

3 citations · 7 across the 13 of their papers we have counts for

collaborators

15 papers

math.CO2022

Towards characterizing the 2-Ramsey equations of the form $a x + b y = p (z)$

Zsolt Baja, Dániel Dobák, Benedek Kovács +2

In this paper, we study a Ramsey-type problem for equations of the form $a x + b y = p (z)$ . We show that if certain technical assumptions hold, then any 2-colouring of the positive integ…

math.CO2022

Irredundant hyperplane covers

János Nagy, Péter Pál Pach, István Tomon

We prove that if $G$ is an abelian group and $H_{1} x_{1}, \dots, H_{k} x_{k}$ is an irredundant (minimal) cover of $G$ with cosets, then $∣ G : i = 1 ⋂ k H_{i} ∣ = 2^{O (k)} .$ This bound…

math.CO2021

Additive bases, coset covers, and non-vanishing linear maps

János Nagy, Péter Pál Pach, István Tomon

Recently, the first two authors proved the Alon-Jaeger-Tarsi conjecture on non-vanishing linear maps, for large primes. We extend their ideas to address several other related conje…

math.CO2021★ 1 cited

The Alon-Jaeger-Tarsi conjecture via group ring identities

János Nagy, Péter Pál Pach

In this paper we resolve the Alon-Jaeger-Tarsi conjecture for sufficiently large primes. Namely, we show that for any finite field $F$ of size $61 < ∣ F ∣ \neq = 79$ and a…

math.CO2020

Avoiding right angles and certain Hamming distances

Balázs Bursics, Dávid Matolcsi, Péter Pál Pach +1

In this paper we show that the largest possible size of a subset of $F_{q}^{n}$ avoiding right angles, that is, distinct vectors $x, y, z$ such that $x - z$ and $y - z$ are perpendi…

math.CO2020★ 1 cited

Sets avoiding six-term arithmetic progressions in $Z_{6}^{n}$ are exponentially small

Péter Pál Pach, Richárd Palincza

We show that sets avoiding 6-term arithmetic progressions in $Z_{6}^{n}$ have size at most $5.70 9^{n}$ . It is also pointed out that the "product construction" does not work in t…

math.CO2020

The counting version of a problem of Erdős

Péter Pál Pach, Richárd Palincza

A set $A$ of natural numbers possesses property $P_{h}$ , if there are no distinct elements $a_{0}, a_{1}, \dots, a_{h} \in A$ with $a_{0}$ dividing the product $a_{1} a_{2} \dots a_{h}$ . Erdő…

math.CO2019

Caps and progression-free sets in $Z_{m}^{n}$

Christian Elsholtz, Péter Pál Pach

We study progression-free sets in the abelian groups $G = (Z_{m}^{n}, +)$ . Let $r_{k} (Z_{m}^{n})$ denote the maximal size of a set $S \subset Z_{m}^{n}$ that does not con…

math.NT2019

On the density of sumsets and product sets

Norbert Hegyvári, François Hennecart, Péter Pál Pach

In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.

math.NT2019★ 3 cited

Coloring the $n$ -smooth numbers with $n$ colors

Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach

For which values of $n$ can we color the positive integers with precisely $n$ colors in such a way that for any $a$ , the numbers $a, 2 a, \dots, na$ all get different colors? Pach pose…

math.NT2018

Polynomial Schur's theorem

Hong Liu, Péter Pál Pach, Csaba Sándor

We resolve the Ramsey problem for ${x, y, z : x + y = p (z)}$ for all polynomials $p$ over $Z$ . In particular, we characterise all polynomials that are $2$ -Ramsey, that is, those…

math.CO2018

The number of maximum primitive sets of integers

Hong Liu, Péter Pál Pach, Richárd Palincza

A set of integers is \emph{primitive} if it does not contain an element dividing another. Denote by $f (n)$ the number of maximum-size primitive subsets of ${1, \dots, 2 n}$ . We pr…

math.NT2018

An improved upper bound for the size of the multiplicative 3-Sidon sets

Péter Pál Pach

We say that a set is a multiplicative 3-Sidon set if the equation $s_{1} s_{2} s_{3} = t_{1} t_{2} t_{3}$ does not have a solution consisting of distinct elements taken from this set. In this paper…

math.NT2016

Multiplicative bases and an Erdős problem

Péter Pál Pach, Csaba Sándor

In this paper we investigate how small the density of a multiplicative basis of order $h$ can be in ${1, 2, \dots, n}$ and in $Z^{+}$ . Furthermore, a related problem of Erd\H…

math.LO2011★ 2 cited

Reducts of the random partial order

Péter Pál Pach, Michael Pinsker, Gabriella Pluhár +2

We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that these structures…