Avoiding Monochromatic Sequences With Special Gaps

For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic sequence $\{x_{1},x_{2},...,x_{k}\}$ with $x_{i}-x_{i-1} \in S$ for $2 \leq i \leq k$. We consider the existence of $f(S,k;r)$ for various choices of $S$, as well as upper and lower bounds on this function. In particular, we show that this function exists for all $k$ if $S$ is an odd translate of the set of primes and $r=2$.Comment: 16 page

Monochromatic paths for the integers

Recall that van der Waerden's theorem states that any finite coloring of the naturals has arbitrarily long monochromatic arithmetic sequences. We explore questions about the set of differences of those sequences

Progressions and Paths in Colorings of Z\mathbb Z

A $\textit{ladder}$ is a set $S \subseteq \mathbb Z^+$ such that any finite coloring of $\mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $\mathbb Z^+$ itself is a ladder. We also discuss variants of ladders, namely $\textit{accessible}$ and $\textit{walkable}$ sets, which are sets $S$ such that any coloring of $\mathbb Z$ contains arbitrarily long (for accessible sets) or infinite (for walkable sets) monochromatic sequences with consecutive differences in $S$. We show that sets with upper density 1 are ladders and walkable. We also show that all directed graphs with infinite chromatic number are accessible, and reduce the bound on the walkability order of sparse sets from 3 to 2, making it tight.Comment: 7 page

New Bounds on van der Waerden-type Numbers for Generalized 3-term Arithmetic Progressions

Let a and b be positive integers with a \leq b. An (a,b)-triple is a set {x,ax+d,bx+ 2d}, where x,d \geq 1. Define T(a,b;r) to be the least positive integer n such that any r-coloring of {1,2...,n} contains a monochromatic (a,b)-triple. Earlier results gave an upper bound on T(a,b;2) that is a fourth degree polynomial in b and a, and a quadratic lower bound. A new upper bound for T(a,b;2) is given that is a quadratic. Additionally, lower bounds are given for the case in which a = b, updated tables are provided, and open questions are presented.Comment: The newer version reflects an updated table of values of T(a,b

On Three Sets with Nondecreasing Diameter

Let $[a,b]$ denote the integers between $a$ and $b$ inclusive and, for a finite subset $X \subseteq \mathbb{Z}$, let the diameter of $X$ be equal to $\max(X)-\min(X)$. We write $X<_p\,Y$ provided $\max(X)<\min(Y)$. For a positive integer $m$, let $f(m,m,m;2)$ be the least integer $N$ such that any $2$-coloring $\Delta: [1, N]\rightarrow \{0,1\}$ has three monochromatic $m$-sets $B_1, B_2, B_3 \subseteq [1,N]$ (not necessarily of the same color) with $B_1<_p\, B_2 <_p\, B_3$ and $diam(B_1)\leq diam(B_2)\leq diam(B_3)$. Improving upon upper and lower bounds of Bialostocki, Erd\H os and Lefmann, we show that $f(m,m,m;2)=8m-5+\lfloor\frac{2m-2}{3}\rfloor+\delta$ for $m\geq 2$, where $\delta=1$ if $m\in \{2,5\}$ and $\delta=0$ otherwise.Comment: 24 page

Avoiding monochromatic sequences with special gaps, preprint

Abstract. For S ⊆ Z + and k and r fixed positive integers, denote by f(S, k; r) the least positive integer n (if it exists) such that within every r-coloring of {1, 2,..., n} there must be a monochromatic sequence {x1, x2,..., xk} with xi − xi−1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S, k; r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if S is an odd translate of the set of primes and r = 2