arXiv subjects
Publications and source records attributed to Aaron Robertson.
We investigate the Hales-Jewett numbers and some variants of them, both computationally and via enumerative and probabilistic arguments. In particular, we give the improved lower bound formula on the combinatorial-geometric variant of the Hales-Jewett numbers. {As a consequence of one of the variants investigated, we show that the Milton Bradley game Connect Four played with 2 players on a 5-dimensional hypercube of any side length can end in a draw.
Schur's Theorem states that, for any $r \in \mathbb{Z}^+$, there exists a minimum integer $S(r)$ such that every $r$-coloring of $\{1,2,\dots,S(r)\}$ admits a monochromatic solution to $x+y=z$. Recently, Budden determined the related Gallai-Schur numbers; that is, he determined the minimum integer $GS(r)$ such that every $r$-coloring of $\{1,2,\dots,GS(r)\}$ admits either a rainbow or monochromatic solution to $x+y=z$. In this article we consider problems that have been solved in the monochromatic setting under a monochromatic-rainbow paradigm. In particular, we investigate Gallai-Schur numbers when $x \neq y$, we consider $x+y+b=z$ and $x+y<z$, and we investigate the asymptotic minimum number of rainbow and monochromatic solutions to $x+y=z$ and $x+y<z$.
We extend Deuber's theorem on $(m,p,c)$-sets to hold over the multidimensional positive integer lattices. This leads to a multidimensional Rado theorem where we are guaranteed monochromatic multidimensional points in all finite colorings of $\left(\mathbb{Z}^+\right)^d$ where the $i^{\mathrm{th}}$ set of coordinates satisfies the $i^{\mathrm{th}}$ given linear Rado system.
We introduce Mitiq, a Python package for error mitigation on noisy quantum computers. Error mitigation techniques can reduce the impact of noise on near-term quantum computers with minimal overhead in quantum resources by relying on a mixture of quantum sampling and classical post-processing techniques. Mitiq is an extensible toolkit of different error mitigation methods, including zero-noise extrapolation, probabilistic error cancellation, and Clifford data regression. The library is designed to be compatible with generic backends and interfaces with different quantum software frameworks. We describe Mitiq using code snippets to demonstrate usage and discuss features and contribution guidelines. We present several examples demonstrating error mitigation on IBM and Rigetti superconducting quantum processors as well as on noisy simulators.
This article documents my journey down the rabbit hole, chasing what I have come to know as a particularly unyielding problem in Ramsey theory on the integers: the $2$-Large Conjecture. This conjecture states that if $D \subseteq \mathbb{Z}^+$ has the property that every $2$-coloring of $\mathbb{Z}^+$ admits arbitrarily long monochromatic arithmetic progressions with common difference from $D$ then the same property holds for any finite number of colors. We hope to provide a roadmap for future researchers and also provide some new results related to the $2$-Large Conjecture.
Consider the equation $\mathcal{E}: x_1+ \cdots+x_{k-1} =x_{k}$ and let $k$ and $r$ be positive integers such that $r\mid k$. The number $S_{\mathfrak{z},2}(k;r)$ is defined to be the least positive integer $t$ such that for any 2-coloring $χ: [1, t] \to \{0, 1\}$ there exists a solution $(\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_k)$ to the equation $\mathcal{E}$ satisfying $\displaystyle \sum_{i=1}^kχ(\hat{x}_i) \equiv 0\pmod{r}$. In a recent paper, the first author posed the question of determining the exact value of $S_{\mathfrak{z}, 2}(k;4)$. In this article, we solve this problem and show, more generally, that $S_{\mathfrak{z}, 2}(k, r)=kr - 2r+1$ for all positive integers $k$ and $r$ with $k>r$ and $r \mid k$.
Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $S_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer $n$ such that every coloring $χ:[1,n] \rightarrow \{0,1,\dots,r-1\}$ admits a solution to $\sum_{i=1}^{k-1} x_i = x_k$ with $\sum_{i=1}^{k} χ(x_i) \equiv 0 \,(\mathrm{mod }\,r)$. We give some formulas and lower bounds for various instances.
Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $w_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $χ:[1,w_{\mathrm{\mathfrak{z}}}(k;r)] \rightarrow \{0,1,\dots,r-1\}$ admits a $k$-term arithmetic progression $a,a+d,\dots,a+(k-1)d$ with $\sum_{j=0}^{k-1} χ(a+jd) \equiv 0 \,(\mathrm{mod }\,r)$. We investigate these numbers as well as a "mixed" monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{\mathrm{\mathfrak{z}}}(k;r)$.
We investigate the distributions of the number of: (1) monochromatic complete subgraphs over edgewise 2-colorings of complete graphs; and (2) monochromatic arithmetic progressions over 2-colorings of intervals, as statistical Ramsey theory questions. We present convincing evidence that both distributions are very well-approximated by the Delaporte distribution.
We show that $\sqrt{k}\cdot r^{k/2}$ is a threshold interval length where, under mild conditions, almost every $r$-coloring of an interval of longer length contains a monochromatic $k$-term arithmetic progression, while almost no $r$-coloring of an interval of shorter length contains a monochromatic $k$-term arithmetic progression.
Pirillo and Varricchio, and independently, Halbeisen and Hungerbuhler considered the following problem, open since 1994: Does there exist an infinite word w over a finite subset of Z such that w contains no two consecutive blocks of the same length and sum? We consider some variations on this problem in the light of van der Waerden's theorem on arithmetic progressions.
There exists a minimum integer $N$ such that any 2-coloring of $\{1,2,...,N\}$ admits a monochromatic solution to $x+y+kz =\ell w$ for $k,\ell \in \mathbb{Z}^+$, where $N$ depends on $k$ and $\ell$. We determine $N$ when $\ell-k \in \{0,1,2,3,4,5\}$, for all $k,\ell$ for which ${1/2}((\ell-k)^2-2)(\ell-k+1)\leq k \leq \ell-4$, as well as for arbitrary $k$ when $\ell=2$.
For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\{1,2,...,n\} \to \{1,2,...,s\}$ admits a $k_i$-term arithmetic progression of color $i$ for some $i$, $1 \leq i \leq s$. In the case when $k_1=k_2=...=k_s=k$ we simply write $w(k;s)$. That such a minimum integer exists follows from van der Waerden's theorem on arithmetic progressions. In the present paper we give a lower bound for $w(k,m)$ for each fixed $m$. We include a table with values of $w(k,3)$ which match this lower bound closely for $5 \leq k \leq 16$. We also give an upper bound for $w(k,4)$, an upper bound for $w(4;s)$, and a lower bound for $w(k;s)$ for an arbitrary fixed $k$. We discuss a number of other functions that are closely related to the van der Waerden function.
Let V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of {1,2,...,n}. We show that (1675/32768) n^2 (1+o(1)) <= V(n) <= (117/2192) n^2(1+o(1)). As a consequence, we find that V(n) is strictly greater than the corresponding number for Schur triples (which is (1/22) n^2 (1+o(1)). Additionally, we disprove the conjecture that V(n) = (1/16) n^2(1+o(1)), as well as a more general conjecture.
We show that for any two linear homogenous equations $\mathcal{E}_0,\mathcal{E}_1$, each with at least three variables and coefficients not all the same sign, any 2-coloring of $\mathbb{Z}^+$ admits monochromatic solutions of color 0 to $\mathcal{E}_0$ or monochromatic solutions of color 1 to $\mathcal{E}_1$. We define the 2-color off-diagonal Rado number $RR(\mathcal{E}_0,\mathcal{E}_1)$ to be the smallest $N$ such that $[1,N]$ must admit such solutions. We determine a lower bound for $RR(\mathcal{E}_0,\mathcal{E}_1)$ in certain cases when each $\mathcal{E}_i$ is of the form $a_1x_1+...+a_nx_n=z$ as well as find the exact value of $RR(\mathcal{E}_0,\mathcal{E}_1)$ when each is of the form $x_1+a_2x_2+...+a_nx_n=z$. We then present a Maple package that determines upper bounds for off-diagonal Rado numbers of a few particular types, and use it to quickly prove two previous results for diagonal Rado numbers.
Let $1 \leq a \leq b$ be integers. A triple of the form $(x,ax+d,bx+2d)$, where $x,d$ are positive integers is called an {\em (a,b)-triple}. The {\em degree of regularity} of the family of all $(a,b)$-triples, denoted dor($a,b)$, is the maximum integer $r$ such that every $r$-coloring of $\mathbb{N}$ admits a monochromatic $(a,b)$-triple. We settle, in the affirmative, the conjecture that dor$(a,b) < \infty$ for all $(a,b) \neq (1,1)$. We also disprove the conjecture that dor($a,b) \in \{1,2,\infty\}$ for all $(a,b)$.
For positive integers $r,k_0,k_1,...,k_{r-1},$ the van der Waerden number $w(k_0,k_1,...,k_{r-1})$ is the least positive integer $n$ such that whenever $\{1,2,...,n\}$ is partitioned into $r$ sets $S_{0},S_{1},...,S_{r-1}$, there is some $i$ so that $S_i$ contains a $k_i$-term arithmetic progression. We find several new exact values of $w(k_0,k_1,...,k_{r-1})$. In addition, for the situation in which only one value of $k_i$ differs from 2, we give a precise formula for the van der Waerden function (provided this one value of $k_i$ is not too small)
A sequence of positive integers $w_1,w_2,...,w_n$ is called an ascending wave if $w_{i+1}-w_i \geq w_i - w_{i-1}$ for $2 \leq i \leq n-1$. For integers $k,r\geq1$, let $AW(k;r)$ be the least positive integer such that under any $r$-coloring of $[1,AW(k;r)]$ there exists a $k$-term monochromatic ascending wave. The existence of $AW(k;r)$ is guaranteed by van der Waerden's theorem on arithmetic progressions since an arithmetic progression is, itself, an ascending wave. Originally, Brown, Erdős, and Freedman defined such sequences and proved that $k^2-k+1\leq AW(k;2) \leq {1/3}(k^3-4k+9)$. Alon and Spencer then showed that $AW(k;2) = O(k^3)$. In this article, we show that $AW(k;3) = O(k^5)$ as well as offer a proof of the existence of $AW(k;r)$ independent of van der Waerden's theorem. Furthermore, we prove that for any $ε> 0$, $$ \frac{k^{2r-1-ε}}{2^{r-1}(40r)^{r^2-1}}(1+o(1)) \leq AW(k;r) \leq \frac{k^{2r-1}}{(2r-1)!}(1+o(1)) $$ holds for all $r \geq 1$, which, in particular, improves upon the best known upper bound for $AW(k;2)$. Additionally, we show that for fixed $k \geq 3$, $$ AW(k;r)\leq\frac{2^{k-2}}{(k-1)!} r^{k-1}(1+o(1)). $$