Trending in Combinatorics
2603.09172Reinforced Generation of Combinatorial Structures: Ramsey Numbers
We present improved lower bounds for five classical Ramsey numbers: $\mathbf{R}(3, 13)$ is increased from $60$ to $61$, $\mathbf{R}(3, 18)$ from $99$ to $100$, $\mathbf{R}(4, 13)$ from $138$ to $139$, $\mathbf{R}(4, 14)$ from $147$ to $148$, and $\mathbf{R}(4, 15)$ from $158$ to $159$. These results were achieved using~\emph{AlphaEvolve}, an LLM-based code mutation agent. Beyond these new results, we successfully recovered lower bounds for all Ramsey numbers known to be exact, and matched the best known lower bounds across many other cases. These include bounds for which previous work does not detail the algorithms used. Virtually all known Ramsey lower bounds are derived computationally, with bespoke search algorithms each delivering a handful of results. AlphaEvolve is a single meta-algorithm yielding search algorithms for all of our results.
2603.09172
Mar 2026Combinatorics
Chromatic thresholds for linear equations and recurrence
Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study $\mathcal L$-solution-free sets $A\subseteq \mathbb F_p$ through the chromatic number of the Cayley graph $\mathsf{Cay}(\mathbb F_p,A)$. We introduce the \emph{chromatic threshold} $δ_χ(\mathcal L)$, the minimum density that guarantees bounded chromatic number of $\mathsf{Cay}(\mathbb F_p,A)$ among all $\mathcal L$-solution-free sets $A$, and determine exactly when $δ_χ(\mathcal L)=0$. We prove that $δ_χ(\mathcal L)=0$ if and only if $\mathcal L$ contains a zero-sum subcollection of at least three coefficients. A key ingredient is a quantitative chromatic lower bound for Cayley graphs on $\mathbb Z_p^n$ generated by Hamming balls around the all-ones vector. This is obtained by introducing a new Kneser-type graph that admits a natural embedding into $\mathbb Z_p^n$, together with an equivariant Borsuk--Ulam type argument. As a consequence, we resolve a question of Griesmer. We further relate our classification to the hierarchy of measurable, topological, and Bohr recurrence. In particular, we show that every infinite discrete abelian group admits a set that is topological recurrent but not measurable recurrent, extending the seminal examples of Kříž and Ruzsa.
2603.05490
Mar 2026Combinatorics
2602.03716Fel's Conjecture on Syzygies of Numerical Semigroups
Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $σ_k=\sum_i d_i^k$ (and $δ_k=(σ_k-1)/2^k$). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture.
2602.03716
Feb 2026Combinatorics
2601.00766Set mappings for general graphs
The study of extremal problems for set mappings has a long history. It was introduced in 1958 by Erdős and Hajnal, who considered the case of cliques in graphs and hypergraphs. Recently, Caro, Patkós, Tuza and Vizer revisited this subject, and initiated the systematic study of set mapping problems for general graphs. In this paper, we prove the following result, which answers one of their questions. Let $G$ be a graph with $m$ edges and no isolated vertices and let $f : E(K_N) \rightarrow E(K_N)$ such that $f(e)$ is disjoint from $e$ for all $e \in E(K_N)$. Then for some absolute constant $C$, as long as $N \geq C m$, there is a copy $G^*$ of $G$ in $K_N$ such that $f(e)$ is disjoint from $V(G^*)$ for all $e \in E(G^*)$. The bound $N = O(m)$ is tight for cliques and is tight up to a logarithmic factor for all $G$.
2601.00766
Jan 2026Combinatorics
Tiling with Boundaries: Dense digital images have large connected components
If most of the pixels in an $n \times m$ digital image are the same color, must the image contain a large connected component? How densely can a given set of connected components pack in $\mathbb{Z}^2$ without touching? We answer these two closely related questions for both 4-connected and 8-connected components. In particular, we use structural arguments to upper bound the "white" pixel density of infinite images whose white (4- or 8-)connected components have size at most $k$. Explicit tilings show that these bounds are tight for at least half of all natural numbers $k$ in the 4-connected case, and for all $k$ in the 8-connected case. We also extend these results to finite images. To obtain the upper bounds, we define the exterior site perimeter of a connected component and then leverage geometric and topological properties of this set to partition images into nontrivial regions called polygonal tiles. Each polygonal tile contains a single white connected component and satisfies a certain maximality property. We then use isoperimetric inequalities to precisely bound the area of these tiles. The solutions to these problems represent new statistics on the connected component distribution of digital images.
2512.11740
Dec 2025Combinatorics
Fault-tolerant mutual-visibility: complexity and solutions for grid-like networks
Networks are often modeled using graphs, and within this setting we introduce the notion of $k$-fault-tolerant mutual visibility. Informally, a set of vertices $X \subseteq V(G)$ in a graph $G$ is a $k$-fault-tolerant mutual-visibility set ($k$-ftmv set) if any two vertices in $X$ are connected by a bundle of $k+1$ shortest paths such that: ($i$) each shortest path contains no other vertex of $X$, and ($ii$) these paths are internally disjoint. The cardinality of a largest $k$-ftmv set is denoted by $\mathrm{f}μ^{k}(G)$. The classical notion of mutual visibility corresponds to the case $k = 0$. This generalized concept is motivated by applications in communication networks, where agents located at vertices must communicate both efficiently (i.e., via shortest paths) and confidentially (i.e., without messages passing through the location of any other agent). The original notion of mutual visibility may fail in unreliable networks, where vertices or links can become unavailable. Several properties of $k$-ftmv sets are established, including a natural relationship between $\mathrm{f}μ^{k}(G)$ and $ω(G)$, as well as a characterization of graphs for which $\mathrm{f}μ^{k}(G)$ is large. It is shown that computing $\mathrm{f}μ^{k}(G)$ is NP-hard for any positive integer $k$, whether $k$ is fixed or not. Exact formulae for $\mathrm{f}μ^{k}(G)$ are derived for several specific graph topologies, including grid-like networks such as cylinders and tori, and for diameter-two networks defined by Hamming graphs and by the direct product of complete graphs.
2512.01978
Dec 2025Combinatorics
