Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.30410

Density regularity of $\{x,x+y,xy\}$ in the integers

Florian K. Richter·Konstantinos Tsinas
View original source ↗Read source document ↗Export metadata ↓

Abstract

Fix $k,s\in\mathbb{N}$. We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in\mathbb{Z}[y]$ of degree $k$ and with $P(0)=0$, every set of positive density contains configurations $\{x,x+P(y),xy^s\}$ for arbitrarily large $x>y\geq 2$. This provides a density strengthening of Moreira's result on partition regularity for $\{x,x+y,xy\}$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Florian K. Richter, Konstantinos Tsinas. 2026-10-07. Density regularity of $\{x,x+y,xy\}$ in the integers. https://arxiv.org/abs/2609.30410

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Upper Clique Transversal Problem: Hardness and Algorithms

A clique transversal in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the study of the ''upper'' variant of this parameter, the upper clique transversal number, defined as the maximum size of a minimal clique transversal. We investigate this parameter from the algorithmic and complexity points of view, with a focus on various graph classes. We show that the corresponding decision problem is NP-complete in the classes of chordal graphs, chordal bipartite graphs, cubic planar bipartite graphs, and line graphs of bipartite graphs, but solvable in linear time in the classes of split graphs, proper interval graphs, and cographs, and in polynomial time for graphs of bounded cliquewidth. We conclude the paper with a number of open questions.

math.CO↗
arXiv2026

Trees with Prescribed Maximum Degree and Spectral Radius

It is well known that the spectral radius $ρ(T)$ of a tree $T$ with at least $3$ vertices satisfies $\frac 14ρ(T)^2+1<Δ(T)\le ρ(T)^2$, where $Δ(T)$ is the maximum degree of $T$. Let $\mathbb{P}$ denote the set of spectral radii of all non-trivial trees. We ask whether, for every $α\in \mathbb{P}\setminus \{1\}$ and every integer $r$ satisfying $\frac 14α^2+1<r\le α^2$, there exists a tree $T$ such that $Δ(T)=r$ and $ρ(T)=α$. For a positive integer $r$ and a positive real number $α$, let ${\mathscr W}_r(α)$ be the smallest subset of $[0,\infty)$ containing $α$ and closed under the following operation: if $q_1,\ldots,q_s$ are positive elements of ${\mathscr W}_r(α)$, not necessarily distinct, and $q=α-\sum_{i=1}^{s}q_i^{-1}$, then $q\in{\mathscr W}_r(α)$ whenever either $1\le s<r$ and $q\ge0$, or $s=r$ and $q=0$. We prove that $0\in {\mathscr W}_r(α)$ if and only if there exists a tree $T$ with $Δ(T)\le r$ and $ρ(T)=α$. Moreover, there exists a tree $T$ with $Δ(T)=r$ and $ρ(T)=α$ if and only if there are positive elements $q_1,\ldots,q_r\in\mathscr W_r(α)$, not necessarily distinct, satisfying $\sum_{i=1}^{r}q_i^{-1}=α$. Consequently, $\mathbb{P}=\{α\geq1:0\in {\mathscr W}_{\lfloorα^2\rfloor}(α)\}.$ As an application, we show that for integers $k,r\ge 2$, there exists a tree $T$ with $Δ(T)=r$ and $ρ(T)=\sqrt k$ if and only if $\frac 14 k+1<r\le k$.

math.CO↗
arXiv2026

Monomial expansions of modified Macdonald polynomials

We discover a family $A$ of statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\widetilde{H}_λ(X;q,t)=\sum_{σ\in T(λ)}x^σt^{maj(σ)}q^{η(σ)}$$ for each statistic $η\in A$. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, $$\widetilde{H}_λ(X;q,t)=\sum_σd_{\varepsilon}(σ)x^σt^{maj(σ)}q^{η(σ)},$$ which are summed over all canonical or dual canonical fillings of a Young diagram, where $d_{\varepsilon}(σ)$ is a product of $q$-multinomials. Finally, the compact formulas enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula due to Garbali and Wheeler (2020), obtained by making extensive use of the Yang-Baxter integrability approach.

math.CO↗