Selection-structure generalizations of the Borsuk-Ulam theorem

Pablo Soberón

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arXiv · 2607.21410

Selection-structure generalizations of the Borsuk-Ulam theorem

Pablo Soberón

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Abstract

We prove Borsuk-Ulam-type results governed by selection structures. Selection structures extend the matroidal framework for colorful theorems in discrete geometry and include non-matroidal examples such as chessboard complexes. Motivated by Frick and Wellner's Radon-type strengthening of Fan's theorem and its colorful variants, we prove selection-structure analogues whose conclusions are determined by Radon partitions. We also prove a prime-power selection-structure covering version of Volovikov's theorem, governed by Tverberg partitions. We include applications to fair partitions, including selection-structure versions of the ham sandwich and necklace splitting theorems.

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Pablo Soberón. 2026-07-23. Selection-structure generalizations of the Borsuk-Ulam theorem. https://arxiv.org/abs/2607.21410

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