Travis Dillon

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20242026

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4 papers · 1 filter

math.CO2025

At most 10 cylinders mutually touch: a Ramsey-theoretic approach

Travis Dillon, Junnosuke Koizumi, Sammy Luo

Littlewood asked for the maximum number $N$ of congruent infinite cylinders that can be arranged in $R^{3}$ so that every pair touches. We improve upon the proof of the seco…

math.CO2025

Improved Helly numbers of product sets

Srinivas Arun, Travis Dillon

A finite family $F$ of convex sets is $k$ -intersecting in $S \subseteq R^{d}$ if the intersection of every subset of $k$ convex sets in $F$ contains a poin…

math.CO2025

Piercing intersecting convex sets

Imre BÃ¡rÃ¡ny, Travis Dillon, DÃ¶mÃ¶tÃ¶r PÃ¡lvÃ¶lgyi +1

Assume two finite families $A$ and $B$ of convex sets in $R^{3}$ have the property that $A \cap B \neq = \emptyset$ for every $A \in A$ and $B\in \math…

math.CO2024

The prime grid contains arbitrarily large empty polygons

Travis Dillon

This paper proves a 2017 conjecture of De Loera, La Haye, Oliveros, and RoldÃ¡n-Pensado that the "prime grid" $\big\{(p,q) \in \mathbb{Z}^2 : \text{$p $an d$ q$ are prime}\big\} \su…