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Crosswords are a form of word puzzle that require a solver to demonstrate a high degree of proficiency in natural language understanding, wordplay, reasoning, and world knowledge, along with adherence to character and length constraints. In this paper we tackle the challenge of solving crosswords with Large Language Models (LLMs). We demonstrate that the current generation of state-of-the art (SoTA) language models show significant competence at deciphering cryptic crossword clues, and outperform previously reported SoTA results by a factor of 2-3 in relevant benchmarks. We also develop a search algorithm that builds off this performance to tackle the problem of solving full crossword grids with LLMs for the very first time, achieving an accuracy of 93\% on New York Times crossword puzzles. Contrary to previous work in this area which concluded that LLMs lag human expert performance significantly, our research suggests this gap is a lot narrower., Comment: Edited to include missing citation

Given a drawing $D$ of a graph $G$, we define the crossing number between any two cycles $C_{1}$ and $C_{2}$ in $D$ to be the number of crossings that involve at least one edge from each of $C_1$ and $C_2$ except the crossings between edges that are common to both cycles. We show that if the crossing number between every two cycles in $G$ is even in a drawing of $G$ on the plane, then there is a planar drawing of $G$. This result can be extended to arbitrary surfaces. We also establish an equivalence between our result and a fundamental result due to Cairns-Nikolayevsky and Pelsmajer-Schaefer-\v{S}tefankovi\v{c}, about drawing graphs on surfaces, and derive the Loebl-Masbaum theorem from it., Comment: Included equivalence with an established result, and derived a previous theorem from the result

Let $\left\{ \mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be an infinite sequence of families of compact connected sets in $\mathbb{R}^{d}$. An infinite sequence of compact connected sets $\left\{ B_{n} \right\}_{n\in \mathbb{N}}$ is called heterochromatic sequence from $\left\{ \mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ if there exists an infinite sequence $\left\{ i_{n} \right\}_{n\in \mathbb{N}}$ of natural numbers satisfying the following two properties: (a) $\{i_{n}\}_{n\in \mathbb{N}}$ is a monotonically increasing sequence, and (b) for all $n \in \mathbb{N}$, we have $B_{n} \in \mathcal{F}_{i_n}$. We show that if every heterochromatic sequence from $\left\{ \mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $d+1$ sets that can be pierced by a single hyperplane then there exists a finite collection $\mathcal{H}$ of hyperplanes from $\mathbb{R}^{d}$ that pierces all but finitely many families from $\left\{ \mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$. As a direct consequence of our result, we get that if every countable subcollection from an infinite family $\mathcal{F}$ of compact connected sets in $\mathbb{R}^{d}$ contains $d+1$ sets that can be pierced by a single hyperplane then $\mathcal{F}$ can be pierced by finitely many hyperplanes. To establish the optimality of our result we show that, for all $d \in \mathbb{N}$, there exists an infinite sequence $\left\{ \mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ of families of compact connected sets satisfying the following two conditions: (1) for all $n \in \mathbb{N}$, $\mathcal{F}_{n}$ is not pierceable by finitely many hyperplanes, and (2) for any $m \in \mathbb{N}$ and every sequence $\left\{B_n\right\}_{n=m}^{\infty}$ of compact connected sets in $\mathbb{R}^d$, where $B_i\in\mathcal{F}_i$ for all $i \geq m$, there exists a hyperplane in $\mathbb{R}^d$ that pierces at least $d+1$ sets in the sequence., Comment: 23 pages, 7 figures

We give necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p,q)$-problem., Comment: 13 pages

We give a generalization of dimension independent Helly Theorem of Adiprasito, B\'{a}r\'{a}ny, Mustafa, and Terpai (Discrete & Computational Geometry 2022) to higher dimensional transversal. We also prove some impossibility results that establish the tightness of our extension., Comment: 10 pages

An infinite sequence of sets $\left\{B_{n}\right\}_{n\in\mathbb{N}}$ is said to be a heterochromatic sequence from an infinite sequence of families $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$, if there exists a strictly increasing sequence of natural numbers $\left\{ i_{n}\right\}_{n \in \mathbb{N}}$ such that for all $n \in \mathbb{N}$ we have $B_{n} \in \mathcal{F}_{i_{n}}$. In this paper, we have proved that if for each $n\in\mathbb{N}$, $\mathcal{F}_n$ is a family of {\em nicely shaped} convex sets in $\mathbb{R}^d$ such that each heterochromatic sequence $\left\{B_{n}\right\}_{n\in\mathbb{N}}$ from $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$ contains at least $k+2$ sets that can be pierced by a single $k$-flat ($k$-dimensional affine space) then all but finitely many families in $\left\{\mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ can be pierced by finitely many $k$-flats. This result can be considered as a {\em countably colorful} generalization of the $(\aleph_0, k+2)$-theorem proved by Keller and Perles (Symposium on Computational Geometry 2022). We have also established the tightness of our result by proving a number of no-go theorems., Comment: 17 pages and 5 figures