AI 中文总结
该研究在凸性空间中证明了五个有界性条件等价，实现了Bukh反例的多项式规模实现，给出了可分凸性空间的首个维度一致的弱埃克霍夫阶特弗贝格界。
AI 中文摘要
赫利数、卡拉西奥多里数和拉东数编码了凸性空间中的三类有限证明：交集为空的证明、属于凸包的证明以及相交包存在的证明。我们研究这些证明的精确版本，其中子族必须保留整个交集，或子集必须保留整个包。我们的第一个主要结果表明，对于任意凸性空间中的有限构型，五个先验不同的有界性条件是等价的：VC维、强赫利数、强卡拉西奥多里数、匹配数和强拉东数（具有预期的加1偏移）。我们还获得了等价的分层特弗贝格型分解和彩色推论，其共同机制由点与生成族之间的二部关联图揭示。对于有限空间，唯一的最小生成元产生了一个自然的对偶凸性空间；我们刻画了双重对偶性并证明强参数是对偶不变的。同一模型给出了Bukh对Calder-Eckhoff划分猜想的反例的多项式规模、O(t⁴)实现。最后，我们获得了可分凸性空间的首个特弗贝格界，该界同时在部分数量上为线性，在拉东数上为多项式。若S₃-可分凸性空间的赫利数为h，且其半空间的VC维为d，则rₜ=O(dh log h)·t；特别地，拉东数r给出rₜ=O(r² log r)·t。当赫利数有界时，该界达到弱埃克霍夫尺度O(rt)。对于ℝᵏ中的轴对齐盒凸性，其在每个维度上均给出最优阶rₜ=O(rt)，这似乎是盒凸性的首个维度一致的弱埃克霍夫阶估计，而此前的直接理论仅局限于三维。
英文摘要
Helly, Carathéodory, and Radon numbers encode three kinds of finite certificates in a convexity space: for the emptiness of an intersection, for membership in a convex hull, and for the existence of intersecting hulls. We study exact versions of these certificates, in which a subfamily must preserve the whole intersection or a subset must preserve the whole hull. Our first main result shows that, for finite configurations in an arbitrary convexity space, five a priori different boundedness conditions are equivalent: VC-dimension, strong Helly number, strong Carathéodory number, comatching number, and strong Radon number (with the expected additive-one shift). We also obtain equivalent layered Tverberg-type decompositions and colorful consequences. The common mechanism is exposed by the bipartite incidence graph between points and a generating family. For finite spaces, the unique minimal generator yields a natural dual convexity space; we characterize double dualization and prove that the strong parameters are duality invariant. The same model gives a polynomial-size, $O(t^4)$, realization of Bukh's counterexample to the Calder-Eckhoff partition conjecture. Finally, we obtain the first Tverberg bound for separable convexity spaces that is simultaneously linear in the number of parts and polynomial in the Radon number. If an $S_3$-separable convexity space has Helly number $h$ and its halfspaces have VC-dimension $d$, then $r_t=O(dh\log h)\,t$; in particular, Radon number $r$ gives $r_t=O(r^2\log r)\,t$. The bound attains the weak-Eckhoff scale $O(rt)$ whenever the Helly number is bounded. For axis-parallel box convexity in $\mathbb{R}^k$, gives the optimal order $r_t=O(rt)$ uniformly in every dimension. This appears to be the first dimension-uniform estimate of weak-Eckhoff order for box convexity, whereas the previous direct theory was confined to dimension three.
Comments23 pages
