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William J. Wesley

Publications and source records attributed to William J. Wesley.

9 recordsLinked to original sources
arXiv2026

Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment

We study autonomous mathematical discovery in the Station, an open-world multi-agent environment in which AI agents from different model families pursue a shared research goal without a central coordinator or scripted pipeline. Agents choose their own research directions, conduct experiments, collaborate and publish papers. These papers accumulate into a shared body of knowledge that later agents can read, cite and extend. We evaluated the Station on 12 mathematical construction problems from the AlphaEvolve study and two additional case studies. Five of the 12 problems yielded results novel relative to the prior literature: a new infinite family of finite field Kakeya sets, new exact 604-point kissing configurations in eleven dimensions, improved bounds for the discretized Kakeya needle and sign uncertainty problems, and a substantially improved lower bound for Erdős's minimum overlap problem. Agents also discovered novel infinite families for Book Ramsey numbers. Their research extended beyond searching for high-scoring constructions: agents developed explanations of their findings and proved theorems outside the assigned tasks. These explanations guided further discoveries and were preserved in the agents' papers, making the underlying insights easier for external researchers to understand and build upon. All presented discoveries are supported by exact constructions or proofs formally verified in Lean. We release the source code, full agent dialogues, papers and verification code, providing a transparent record of how these discoveries emerged.

cs.AI↗
arXiv2026

A lower bound on the Ramsey number $R_k(k+1,k+1)$

We will prove that $R_k(k+1,k+1)\geq 4 tw_{\lfloor k/4\rfloor -3}(2)$, where $tw$ is the tower function defined by ${tw}_1(x)=x$ and ${tw}_{i+1}(x)=2^{{tw}_i(x)}$. We also give proofs of $R_k(k+1,k+2)\geq 4 tw_{k-7}(2)$, $R_k(k+1,2k+1)\geq 4 tw_{k-3}(2)$, and $R_k(k+2,k+2)\geq 4 tw_{k-4}(2)$.

math.CO↗
arXiv2025

Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellensätze

In this article we study the Ramsey numbers $R(r,s)$ through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions correspond to Ramsey graphs of order $n$, those that do not contain a copy of $K_r$ or $\bar{K}_s$. When these systems have no solution and $n \ge R(r,s)$, we construct Nullstellensatz certificates whose degrees are equal to the restricted online Ramsey numbers introduced by Conlon, Fox, Grinshpun and He. Moreover, we show that these results generalize to other numbers in Ramsey theory, including Rado, van der Waerden, and Hales-Jewett numbers. Finally, we introduce a family of numbers that relate to the coefficients of a certain "Ramsey polynomial" that gives lower bounds for Ramsey numbers.

math.CO↗
arXiv2025

New bounds for some small multicolor Ramsey numbers

The Ramsey number $R(G_1,\dots,G_k)$ is the smallest $n$ such that every $k$-coloring of the edges of $K_n$ contains a monochromatic copy of $G_i$ in color $i$. Ramsey numbers are challenging to compute, and few are known exactly. We use Boolean satisfiability (SAT) solvers to search for structured colorings that give lower bounds, and we show $R(K_4,K_4-e,K_4-e) \ge 35$ and $R(K_3,K_4,C_4,C_4) \ge 49$. Moreover, we tighten some recent upper bounds for multicolor Ramsey numbers for cycles and show $R(C_3,C_6,C_6) = R(C_5,C_6,C_6) = 15$. Finally, we enumerate critical graphs for the numbers $R(C_4,K_{1,s})$ and $R(C_6,K_{1,s})$.

math.CO↗
arXiv2025

Lower Bounds for Book Ramsey Numbers

We prove new bounds for Ramsey numbers for book graphs $B_n$. In particular, we show that $R(B_{n-1},B_n) = 4n-1$ for an infinite family of $n$ using a block-circulant construction similar to Paley graphs. We obtain improved bounds for several other values of $R(B_r,B_s)$ using different block-circulant graphs from SAT and integer programming (IP) solvers. Finally, we enumerate the number of critical graphs for $R(B_r,B_s)$ for small $r$ and $s$ using SAT modulo symmetries (SMS).

math.CO↗
arXiv2024

Optimization Tools for Computing Colorings of $[1,\cdots ,n]$ with Few Monochromatic Solutions on $3$-variable Linear Equations

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge R_k(E)$, there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of $n$? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.

math.CO↗
arXiv2022

Improved Ramsey-type theorems for Fibonacci numbers and other sequences

Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression. We consider two variants of van der Waerden numbers: the numbers $n = n(AP_D,k;r)$, the smallest value where every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression with common difference in $D$, and the numbers $n = Δ(D,k;r)$, the smallest value $n$ where every $r$-coloring of $\{1,\dots,n\}$ contains a sequence $x_1 < \dots < x_k$ where the differences between consecutive terms are members of $D$. We study the case when $D$ is set of Fibonacci numbers $F$ and give improved bounds for the largest $r$ where $n(AP_F,k;r)$ and $Δ(F,k;r)$ exist for all $k$. Moreover, we give some computational data on $Δ(D,k;r)$ for other sets $D$.

math.CO↗
arXiv2022

Rado Numbers and SAT Computations

Given a linear equation $\mathcal{E}$, the $k$-color Rado number $R_k(\mathcal{E})$ is the smallest integer $n$ such that every $k$-coloring of $\{1,2,3,\dots,n\}$ contains a monochromatic solution to $\mathcal E$. The degree of regularity of $\mathcal E$, denoted $dor(\mathcal E)$, is the largest value $k$ such that $R_k(\mathcal E)$ is finite. In this article we present new theoretical and computational results about the Rado numbers $R_3(\mathcal{E})$ and the degree of regularity of three-variable equations $\mathcal{E}$. % We use SAT solvers to compute many new values of the three-color Rado numbers $R_3(ax+by+cz = 0)$ for fixed integers $a,b,$ and $c$. We also give a SAT-based method to compute infinite families of these numbers. In particular, we show that the value of $R_3(x-y = (m-2) z)$ is equal to $m^3-m^2-m-1$ for $m\ge 3$. This resolves a conjecture of Myers and implies the conjecture that the generalized Schur numbers $S(m,3) = R_3(x_1+x_2 + \dots x_{m-1} = x_m)$ equal $m^3-m^2-m-1$ for $m\ge 3$. Our SAT solver computations, combined with our new combinatorial results, give improved bounds on $dor(ax+by = cz)$ and exact values for $1\le a,b,c\le 5 $. We also give counterexamples to a conjecture of Golowich.

math.CO↗