Tverberg cores and Kalai's cascade conjecture
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917516856524800 |
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| author | Soberón, Pablo |
| author_facet | Soberón, Pablo |
| contents | We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an $n$-simplex to $\mathbb R^d$, let $T_r(f)$ be the set of points contained in the images of $r$ pairwise disjoint faces. We prove that if $r$ is a prime power and $\dim T_r(f)\le k$, then there exists a point that remains an $r$-Tverberg point after any $t$ vertices are deleted, provided $n=(r-1)(d+1)+t(k+1)$. For $t=1$, this gives a topological analogue of a standard consequence of Kalai's cascade conjecture. We also confirm the cascade conjecture for finite point sets whose Radon set is $0$-dimensional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21305 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tverberg cores and Kalai's cascade conjecture Soberón, Pablo Combinatorics Algebraic Topology 52A37, 55M20, 55R40, 55R91 We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an $n$-simplex to $\mathbb R^d$, let $T_r(f)$ be the set of points contained in the images of $r$ pairwise disjoint faces. We prove that if $r$ is a prime power and $\dim T_r(f)\le k$, then there exists a point that remains an $r$-Tverberg point after any $t$ vertices are deleted, provided $n=(r-1)(d+1)+t(k+1)$. For $t=1$, this gives a topological analogue of a standard consequence of Kalai's cascade conjecture. We also confirm the cascade conjecture for finite point sets whose Radon set is $0$-dimensional. |
| title | Tverberg cores and Kalai's cascade conjecture |
| topic | Combinatorics Algebraic Topology 52A37, 55M20, 55R40, 55R91 |
| url | https://arxiv.org/abs/2605.21305 |