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arxiv: 2605.21305 · v1 · pith:NMOEENRBnew · submitted 2026-05-20 · 🧮 math.CO · math.AT

Tverberg cores and Kalai's cascade conjecture

Pablo Sober\'on This is my paper

Pith reviewed 2026-05-21 03:08 UTC · model grok-4.3

classification 🧮 math.CO math.AT
keywords Tverberg theoremKalai cascade conjecturetopological combinatoricsr-Tverberg pointssimplex mapsdimension boundsRadon setsprime power
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The pith

For prime-power r, low-dimensional T_r(f) guarantees an r-Tverberg point that survives deletion of any t vertices when n equals (r-1)(d+1) plus t(k+1).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a topological analogue of Kalai's cascade conjecture for maps from an n-simplex to R^d. If r is a prime power and the set T_r(f) of points that lie in r pairwise disjoint faces has dimension at most k, then some point in T_r(f) remains an r-Tverberg point after any t vertices are removed, provided n satisfies the stated formula. This adds a deletion-robustness property to the standard topological Tverberg theorem. The same conclusion holds for finite point sets whose Radon set is zero-dimensional.

Core claim

If r is a prime power and dim T_r(f) ≤ 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 recovers a topological version of a known consequence of Kalai's cascade conjecture, and the cascade conjecture itself is verified when the Radon set of a finite point configuration is zero-dimensional.

What carries the argument

T_r(f), the set of points in R^d that belong to the images of r pairwise disjoint faces of the simplex under the continuous map f; the argument uses its dimension bound to produce a deletion-resistant element inside it.

If this is right

  • For t=1 the result supplies a topological analogue of a standard consequence of Kalai's cascade conjecture.
  • The cascade conjecture holds for all finite point sets whose Radon set is zero-dimensional.
  • Dimension of the Tverberg set controls robustness to vertex deletions in the topological setting.
  • The formula n=(r-1)(d+1)+t(k+1) gives the precise size needed for the robustness statement to hold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar dimension-to-robustness statements may exist for other partition problems such as the colorful Tverberg theorem or the necklace splitting problem.
  • If the result extends beyond prime powers it would resolve the full cascade conjecture in the topological category.
  • The same deletion-robust core construction could be tested numerically on random maps or on convex position point sets in low dimensions.

Load-bearing premise

The topological Tverberg theorem is assumed to hold for prime-power r so that r-Tverberg points are known to exist before the dimension condition on T_r(f) is applied.

What would settle it

A continuous map f from an n-simplex with n=(r-1)(d+1)+t(k+1) to R^d such that dim T_r(f) ≤ k yet every point in T_r(f) ceases to be an r-Tverberg point after some set of t vertices is removed.

Figures

Figures reproduced from arXiv: 2605.21305 by Pablo Sober\'on.

Figure 1
Figure 1. Figure 1: To show that the Radon core is non-empty, we will show the following fact. If we map a (k + 1)-dimensional sphere to a k-dimensional space using a continuous even map Φ, there will be a point p whose preimage connects two antipodal points of the sphere. To distinguish ordered Radon partitions of S, we take X to be the ℓ1-sphere of kerL of radius 2. Those are the non-zero vectors α = (α1, . . . , αn+1) ∈ ke… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies topological analogues of Kalai's cascade conjecture. For a continuous map f from an n-simplex to R^d, let T_r(f) be the set of r-Tverberg points (points in the images of r pairwise disjoint faces). The main theorem states that if r is a prime power and dim T_r(f) ≤ k, then there exists a point in T_r(f) that remains an r-Tverberg point after deletion of any t vertices, provided n = (r-1)(d+1) + t(k+1). For t=1 this yields a topological analogue of a standard consequence of Kalai's conjecture. The authors also confirm the cascade conjecture for finite point sets whose Radon set is 0-dimensional.

Significance. If the central result holds, it supplies a quantitative, dimension-based robustness statement for Tverberg points that directly extends the classical topological Tverberg theorem (valid for prime-power r) to a cascade setting. The approach uses a standard dimension-counting selection argument once an initial non-empty T_r(f) is guaranteed by the background theorem, together with the extra t(k+1) vertices. The 0-dimensional confirmation provides concrete supporting evidence in a special case. These features strengthen the link between equivariant topology and deletion-stable configurations in discrete geometry.

minor comments (3)
  1. §1 (Introduction): the precise definition of the Radon set for the 0-dimensional confirmation case should be restated explicitly, even if it is standard, to make the verification self-contained.
  2. The dimension formula in the main theorem is stated cleanly, but a short remark clarifying why the bound is sharp (or at least why the coefficient t(k+1) is optimal under the given hypotheses) would strengthen the presentation.
  3. A few references to recent work on quantitative Tverberg theorems or cascade-type results in the literature could be added for context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our results on topological analogues of Kalai's cascade conjecture, including the dimension-based robustness statement for Tverberg points when r is a prime power and the confirmation in the 0-dimensional Radon case. We appreciate the recognition that the main theorem extends the classical topological Tverberg theorem in a quantitative way and the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained against external benchmarks

full rationale

The central claim invokes the established topological Tverberg theorem (valid for prime-power r) solely to guarantee a non-empty T_r(f) at the base dimension n=(r-1)(d+1). This is an independent, externally verified result from prior literature, not derived or fitted within the paper. The subsequent dimension-counting argument that selects a point surviving t deletions under the hypothesis dim T_r(f) ≤ k is a standard selection step that does not reduce to any self-definition, fitted input renamed as prediction, or self-citation chain. The confirmation of Kalai's cascade conjecture for 0-dimensional Radon sets is likewise a new, non-circular statement. No load-bearing step collapses by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard topological Tverberg theorem for prime-power r, continuity of the given map, and basic dimension theory for simplicial complexes; no free parameters or invented entities are introduced in the abstract statements.

axioms (2)
  • standard math The topological Tverberg theorem holds for maps from simplices when r is a prime power.
    Invoked implicitly to guarantee existence of initial r-Tverberg points before applying the new dimension condition.
  • domain assumption Continuous maps from simplices to Euclidean space induce well-defined intersection sets T_r(f) whose dimension can be controlled.
    Central to the statement of the main theorem.

pith-pipeline@v0.9.0 · 5649 in / 1459 out tokens · 35657 ms · 2026-05-21T03:08:48.663929+00:00 · methodology

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Reference graph

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5 extracted references · 5 canonical work pages

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