Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.30711

Complete Determination of the Gallai--van der Waerden Number \(\mathrm{GW}(r;3,3)\)

Yuan Si
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper completely determines the Gallai--van der Waerden number \(\mathrm{GW}(r;3,3)\) for all integers \(r\ge 3\): \[ \mathrm{GW}(3;3,3)=\mathrm{GW}(4;3,3)=9,\qquad \mathrm{GW}(r;3,3)=r\quad (r\ge 5). \]

Explore related subjects

math.NTmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuan Si. 2026-10-22. Complete Determination of the Gallai--van der Waerden Number \(\mathrm{GW}(r;3,3)\). https://arxiv.org/abs/2610.30711

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

KEEP EXPLORING

Related papers

arXiv2026

The Recursive Behavior of the Number of Irreducible Factors of Certain Rational Transformations over Finite Fields

Constructing irreducible polynomials over finite fields via rational transformations is an active field of study. In this paper, we show that the method can be used for more than that. By considering normalized rational functions $Q$ that generate an invariant subfield $\mathbb{F}_q(x)^G$ of $\mathbb{F}_q(x)$, for a subgroup $G\le Aut(\mathbb{F}_q(x)/\mathbb{F}_q)$, and exploiting the factorization pattern of their rational transformations, we are able to prove that the sets arising from considering whether the rational transformation $f^Q$ has exactly $deg(Q)$ irreducible factors of the same degree as $f$, or an irreducible factor of higher degree, admit a recursive behavior, i.e. their numbers satisfy a recursive equation. These sets occasionally can be described in terms of arithmetical properties the roots of the irreducible polynomials satisfy, e.g. their roots satisfy certain trace or power/non-power conditions. These can often be translated to properties of their minimal polynomials. With that, we prove a new theorem about the number of certain irreducible factors of a class of linearized polynomials, which gives a new context to a recent paper by Granger.

math.NT↗
arXiv2026

Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions

Fix a Dirichlet character $χ$ and a cuspidal GL$(2)$ eigenform $ϕ$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $π$ on GL$(3)$ such that $L(1/2, π\times χ)$ and $L(1/2, π\times ϕ)$ are simultaneously non-zero. We achieve this by use of Jacquet's Relative Trace Formula. We derive an expression of the average over the GL$(3)$ cuspidal spectrum as a sum of a non-zero main term and two subsidiary terms which are forced to be zero for large enough level by use of a suitable test function. This modest article is dedicated to the memory of Harish Chandra, on the occasion of his hundredth birthday.

math.NT↗
arXiv2026

On the Kummer pro-étale cohomology of $\mathbb B_{\operatorname{dR}}$

We investigate $p$-adic cohomologies of log rigid analytic varieties over a $p$-adic field. For a log rigid analytic variety $X$ defined over a discretely valued field, we compute the Kummer pro-étale cohomology of $\mathbb{B}_{\mathrm{dR}}^+$ and $\mathbb{B}_{\mathrm{dR}}$. When $X$ is defined over $\mathbb{C}_p$, we introduce a logarithmic ${B}_{\mathrm{dR}}^+$-cohomology theory, serving as a deformation of log de Rham cohomology. Additionally, we establish the log de Rham-étale comparison in this setting and prove the degeneration of both the Hodge-Tate and Hodge-log de Rham spectral sequences when $X$ is proper and log smooth.

math.NT↗