On the degree of regularity of a certain quadratic Diophantine equation

We show that, for every positive integer r, there exists an integer b = b(r) such that the 4-variable quadratic Diophantine equation (x1 − y1)(x2 − y2) = b is r-regular. Our proof uses Szemerédi’s theorem on arithmetic progressions

All finite sets are Ramsey in the maximum norm

For two metric spaces $\mathcal X$ and $\mathcal Y$, the chromatic number $\chi(\mathcal X;\mathcal Y)$ of $\mathcal X$ with forbidden $\mathcal Y$ is the smallest $k$ such that there is a coloring of the points of $\mathcal X$ with no monochromatic copy of $\mathcal Y$. In this paper, we show that for each finite metric space $\mathcal{M}$ the value $\chi\left( {\mathbb R}^n_\infty; \mathcal M \right)$ grows exponentially with $n$. We also provide explicit lower and upper bounds for some special $\mathcal M$

The Slice Rank Polynomial Method

Suppose you wanted to bound the maximum size of a set in which every k-tuple of elements satisfied a specific condition. How would you go about this? Introduced in 2016 by Terence Tao, the slice rank polynomial method is a recently developed approach to solving problems in extremal combinatorics using linear algebraic tools. We provide the necessary background to understand this method, as well as some applications. Finally, we investigate a generalization of the slice rank, the partition rank introduced by Eric Naslund in 2020, along with various discussions on the intuition behind the slice rank polynomial method and other possible avenues for generalization

Equation-regular sets and the Fox–Kleitman conjecture

Given k ≥ 1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2k-variable linear Diophantine equation ∑k i=1 (xi − yi) = b is (2k − 1)-regular. This is best possible, since Fox and Kleitman showed that for all b ≥ 1, this equation is not 2k-regular. While the conjecture has recently been settled for all k ≥ 2, here we focus on the case k = 3 and determine the degree of regularity of the corresponding equation for all b ≥ 1. In particular, this independently confirms the conjecture for k = 3. We also briefly discuss the case k = 4

Irredundant sets, Ramsey numbers, multicolor Ramsey numbers

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ that is adjacent to $x$ and to no other vertex of $X$. The \emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in a blue subgraph or a $n$-element independent set in a red subgraph. The \emph{multicolor irredundant Ramsey number} $s(t_{1},\ldots,t_{l})$ is the minimum $r$ such that every $l$-coloring of the edges of the complete graph $K_{r}$ on $r$ vertices has a monochromatic irredundant set of size $s_{i}$ for certain $1\leq i\leq l$. Firstly, we improve the upper bound for the mixed Ramsey number $t(3,n)$, and using this result, we verify a special case of a conjecture proposed by Chen, Hattingh, and Rousseau for $m=4$. Secondly, we obtain a new upper bound for $s(3,9)$, and using Krivelevich's method, we establish an asymptotic lower bound for CO-irredundant Ramsey number of $K_{N}$, which extends Krivelevich's result on $s(m,n)$. Thirdly, we prove a lower bound for the multicolor irredundant Ramsey number by a random and probability method which has been used to improve the lower bound for multicolor Ramsey numbers. Finally, we give a lower bound for the irredundant multiplicity.Comment: 23 pages, 1 figur