Let $$\mathcal {F}$$ F be a family of convex sets in $${\mathbb {R}}^d,$$ R d , which are colored with $$d+1$$ d + 1 colors. We say that $$\mathcal {F}$$ F satisfies the Colorful Helly Property if every rainbow selection of $$d+1$$ d + 1 sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family $$\mathcal {F}$$ F there is a color class $$\mathcal {F}_i\subset \mathcal {F},$$ F i ⊂ F , for $$1\le i\le d+1,$$ 1 ≤ i ≤ d + 1 , whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension $$d\ge 2$$ d ≥ 2 there exist numbers f ( d ) and g ( d ) with the following property: either one can find an additional color class whose sets can be pierced by f ( d ) points, or all the sets in $$\mathcal {F}$$ F can be crossed by g ( d ) lines.

中文翻译：

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多彩地狱假说的进一步后果

令 $$\mathcal {F}$$ F 是 $${\mathbb {R}}^d,$$ R d 中的一组凸集，它们用 $$d+1$$ d + 1 种颜色着色. 如果 $$d+1$$ d + 1 个集合中的每个彩虹选择（每个颜色类中的一个集合）都有一个非空的公共交集，我们说 $$\mathcal {F}$$ F 满足彩色 Helly 属性。Lovász 的多彩 Helly 定理指出，对于任何这样的多彩族 $$\mathcal {F}$$ F，有一个颜色类 $$\mathcal {F}_i\subset \mathcal {F},$$ F i ⊂ F , 对于 $$1\le i\le d+1,$$ 1 ≤ i ≤ d + 1 ，其集合具有非空交集。我们建立了五颜六色的海莉假说的进一步后果。特别地，我们表明对于每个维度 $$d\ge 2$$ d ≥ 2 存在具有以下属性的数字 f ( d ) 和 g ( d )：任何一个都可以找到一个额外的颜色类，其集合可以被刺穿通过 f ( d ) 点，