arXiv · 2609.37159
We prove the equivalence of Hindman's finite sums theorem with a natural extension of Ramsey's theorem.
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Lorenzo Carlucci, Andrea D'Amico. 2026-09-29. An equivalent form of Hindman's Theorem. https://arxiv.org/abs/2609.37159
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Let $x_0(q)=-ξ_0(q)$ be the leading formal root of $Θ_0(x,q)=\sum_{n\geq0}x^nq^{\binom n2}$. I give here explicit combinatorial interpretations of the positive integer coefficients of $ξ_0(q)=1+q+2q^2+4q^3+9q^4+\cdots$ in terms of rooted trees enriched by stack polyominoes or certain Ferrers diagrams, weighted by total area. The two enrichments may be chosen independently at each level of the tree. A decomposition along the first-child path gives a combinatorial interpretation of $1-ξ_0^{-1}$. By reserving two successor slots at the root, I also obtain an interpretation of $1-ξ_0^{-2}$ and its zero coefficient in degree three. The sequence decomposition gives a Lyndon-word interpretation of the Euler-product exponents and proves their positivity and weak monotonicity. Finally, I derive the coefficient asymptotic $[q^n]ξ_0(q)\sim ξ_0(ρ)ρ^{-n}n^{-3/2}/(2\sqrtπ)$, where $ρ$ is the radius of convergence. The tree models are equinumerous with the braid classes studied by Flores and González-Meneses.
The combinatorial mutation $\mathrm{mut}_w(P,F)$ for a lattice polytope $P$ was introduced in the context of mirror symmetry for Fano manifolds in [1]. It was also proved in \cite{ACGK} that for a lattice polytope $P \subseteq N_\mathbb{R}$ containing the origin in its interior, the polar dual $P^* \subseteq M_\mathbb{R}$ and $\mathrm{mut}_w(P,F)^* \subseteq M_\mathbb{R}$ have the same Ehrhart quasi-polynomial. To extend this framework, we introduce combinatorial mutation for rational pointed polyhedra in $N_\mathbb{R}$ containing the origin in their interiors. Such polyhedra are Minkowski sums of rational polytopes and rational polyhedral pointed cones. On the dual side $M_\mathbb{R}$, the construction applies to full-dimensional rational polytopes containing the origin, not necessarily in their interiors. As an application of this extension of the combinatorial mutation, we prove that the chain polytope of a poset $Π$ can be obtained by a sequence of combinatorial mutations in $M_\mathbb{R}$ from the order polytope of $Π$. Namely, the order polytope and the chain polytope of the same poset $Π$ are mutation-equivalent.
We study a variant of the Cops and Robbers game on graphs in which the robbers damage the visited vertices, aiming to maximize the number of damaged vertices. For that game with one cop against $s$ robbers a conjecture was made by Carlson, Halloran and Reinhart that the cop can save three vertices from being damaged as soon as the maximum degree of the base graph is at least $\binom{s}{2} + 2$. We are able to verify the conjecture and prove that it is tight once we add the assumption that the base graph is triangle free. We also study the game without that assumption, disproving the conjecture in full generality and further attempting to locate the smallest maximum degree of a base graph which guarantees that the cop can save three vertices against $s$ robbers. We show that this number is between $2\binom{s}{2} - 3$ and $2\binom{s}{2} + 1$. Furthermore, after the game has been previously studied with one cop and multiple robbers, as well as with one robber and multiple cops, we initiate the study of the game with two cops and two robbers. In the case when the base graph is a cycle we determine the exact number of damaged vertices. Additionally, when the base graph is a path we provide bounds that differ by an additive constant.