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Péter Pál Pach

Publications and source records attributed to Péter Pál Pach.

30 records · Page 2Linked to original sources
arXiv2020

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

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 $a,2a,\dots,na$ all get different colors? Pach posed the question around 2008-9. Particular cases appeared in KöMaL in April 2010, and the general version appeared in May 2010 on MathOverflow, posted by Pálvölgyi. The question remains open. We discuss the known partial results and investigate a series of related matters attempting to understand the structure of these $n$-satisfactory colorings. Specifically, we show that there is an $n$-satisfactory coloring whenever there is an abelian group operation $\oplus$ on the set $\{1,2,\dots,n\}$ compatible with multiplication in the sense that whenever $i$, $j$ and $ij$ are in $\{1,\dots,n\}$, then $ij=i\oplus j$. This includes in particular the cases where $n+1$ is prime, or $2n+1$ is prime, or $n=p^2-p$ for some prime $p$, or there is a $k$ such that $q=nk+1$ is prime and $1^k,\dots,n^k$ are all distinct modulo $q$ (in which case we call $q$ a strong representative of order $n$). The colorings obtained by this process we call multiplicative. We also show that nonmultiplicative colorings exist for some values of $n$. There is an $n$-satisfactory coloring of $\mathbb Z^+$ if and only if there is such a coloring of the set $K_n$ of $n$-smooth numbers. We identify all $n$-satisfactory colorings for $n\le 5$ and all multiplicative colorings for $n\le 8$, and show that there are as many nonmultiplicative colorings of $K_n$ as there are real numbers for $n=6$ and 8. We show that if $n$ admits a strong representative $q$ then the set of such $q$ has positive natural density in the set of all primes. We show that the question of whether there is an $n$-satisfactory coloring is equivalent to a problem about tilings, and use this to give a geometric characterization of multiplicative colorings.

math.NT↗
arXiv2019

The step Sidorenko property and non-norming edge-transitive graphs

Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a quasirandom multipartite graph minimizes the density of H among all graphs with the same edge densities between its parts; this property is called the step Sidorenko property. We show that many bipartite graphs fail to have the step Sidorenko property and use our results to show the existence of a bipartite edge-transitive graph that is not weakly norming; this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150].

math.CO↗
arXiv2019

Caps and progression-free sets in $\mathbb{Z}_m^n$

We study progression-free sets in the abelian groups $G=(\mathbb{Z}_m^n,+)$. Let $r_k(\mathbb{Z}_m^n)$ denote the maximal size of a set $S \subset \mathbb{Z}_m^n$ that does not contain a proper arithmetic progression of length $k$. We give lower bound constructions, which e.g. include that $r_3(\mathbb{Z}_m^n) \geq C_m \frac{((m+2)/2)^n}{\sqrt{n}}$, when $m$ is even. When $m=4$ this is of order at least $3^n/\sqrt{n}\gg \vert G \vert^{0.7924}$. Moreover, if the progression-free set $S\subset \mathbb{Z}_4^n$ satisfies a technical condition, which dominates the problem at least in low dimension, then $|S|\leq 3^n$ holds. We present a number of new methods which cover lower bounds for several infinite families of parameters $m,k,n$, which includes for example: $r_6(\mathbb{Z}_{125}^n) \geq (85-o(1))^n$. For $r_3(\mathbb{Z}_4^n)$ we determine the exact values, when $n \leq 5$, e.g. $r_3(\mathbb{Z}_4^5)=124$, and for $r_4(\mathbb{Z}_4^n)$ we determine the exact values, when $n \leq 4$, e.g. $r_4(\mathbb{Z}_4^4)=128$.

math.CO↗
arXiv2018

The number of multiplicative Sidon sets of integers

A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erdős in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the lower order term: $π(n)+Θ(\frac{n^{3/4}}{(\log n)^{3/2}})$. We show that the number of multiplicative Sidon subsets of $\{1,\ldots, n\}$ is $T(n)\cdot 2^{Θ(\frac{n^{3/4}}{(\log n)^{3/2}})}$ for a certain function $T(n)\approx 2^{1.815π(n)}$ which we specify. This is a rare example in which the order of magnitude of the lower order term in the exponent is determined. It resolves the enumeration problem for multiplicative Sidon sets initiated by Cameron and Erdős in the 80s. We also investigate its extension for generalised multiplicative Sidon sets. Denote by $S_k$, $k\ge 2$, the number of multiplicative $k$-Sidon subsets of $\{1,\ldots, n\}$. We show that $S_k(n)=(β_k+o(1))^{π(n)}$ for some $β_k$ we define explicitly. Our proof is elementary.

math.NT↗
arXiv2018

The number of maximum primitive sets of integers

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 $\{1,\ldots, 2n\}$. We prove that the limit $α=\lim_{n\rightarrow \infty}f(n)^{1/n}$ exists. Furthermore, we present an algorithm approximating $α$ with $(1+\varepsilon)$ multiplicative error in $N(\varepsilon)$ steps, showing in particular that $α\approx 1.318$. Our algorithm can be adapted to estimate also the number of all primitive sets in $\{1,\ldots, n\}$. We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in $\{1,\ldots, n\}$ is between $2^{π(n)}\cdot e^{(\frac{1}{2}+o(1))\sqrt{n}}$ and $2^{π(n)}\cdot e^{(2+o(1))\sqrt{n}}$. We show that neither of these bounds is tight: there are in fact $2^{π(n)}\cdot e^{(1+o(1))\sqrt{n}}$ such sets.

math.CO↗
arXiv2018

Monochromatic solutions to $x+y=z^2$ in the interval $[N,cN^4]$

Green and Lindqvist proved that for any 2-colouring of $\mathbb{N}$, there are in\-fi\-ni\-tely many monochromatic solutions to $x+y=z^2$. In fact, they showed the existence of a monochromatic solution in every interval $[N,cN^8]$ with large enough $N$. In this short note we give a different proof for their theorem and prove that a monochromatic solution exists in every interval $[N,10^4N^4]$ with large enough $N$. A 2-colouring of $[N,(1/27)N^4]$ avoiding monochromatic solutions to $x+y=z^2$ is also presented, which shows that in $10^4N^4$ only the constant factor can be reduced.

math.NT↗
arXiv2018

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

We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots,n\}$ is at most $π(n)+π(n/2)+n^{2/3}(\log n )^{2^{1/3}-1/3+o(1)}$, which improves the previously known best bound $π(n)+π(n/2)+cn^{2/3}\log n/\log\log n$.

math.NT↗
arXiv2017

On infinite multiplicative Sidon sets

We prove that if $A$ is an infinite multiplicative Sidon set, then $\liminf\limits_{n\to \infty}\frac{|A(n)|-π(n)}{\frac{n^{3/4}}{(\log n)^3}}<\infty$ and construct an infinite multiplicative Sidon set satisfying $\liminf\limits_{n\to \infty}\frac{|A(n)|-π(n)}{\frac{n^{3/4}}{(\log n)^3}}>0$.

math.NT↗
arXiv2016

Multiplicative bases and an Erdős problem

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 $\mathbb{Z}^+$. Furthermore, a related problem of Erd\H os is also studied: How dense can a set of integers be, if none of them divides the product of $h$ others?

math.NT↗
arXiv2012

A new operation on partially ordered sets

Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and the second type are permutations induced by so-called rotations. In this paper we introduce rotations for finite posets, which can be seen as the poset counterpart of Seidel-switch for finite graphs. We analyze some of their combinatorial properties, and investigate in particular the question of when two finite posets are rotation-equivalent. We moreover give an explicit combinatorial construction of a rotation of the random poset whose image is again isomorphic to the random poset. As an corollary of our results on rotations of finite posets, we obtain that the group of rotating permutations of the random poset is the automorphism group of a homogeneous structure in a finite language.

math.CO↗
arXiv2012

Reducts of the random partial order

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 fall into precisely five equivalence classes. We achieve this result by showing that there exist exactly five closed permutation groups which contain the automorphism group of the random partial order, and thus expose all symmetries of this structure. Our classification lines up with previous similar classifications, such as the structures definable in the random graph or the order of the rationals; it also provides further evidence for a conjecture due to Simon Thomas which states that the number of structures definable in a homogeneous structure in a finite relational language is, up to first-order interdefinability, always finite. The method we employ is based on a Ramsey-theoretic analysis of functions acting on the random partial order, which allows us to find patterns in such functions and make them accessible to finite combinatorial arguments.

math.LO↗