Let $${\mathcal {A}}=\{A_1,\ldots ,A_n\}$$ A = { A 1 , … , A n } be a family of sets in the plane. For $$0 \le i < n$$ 0 ≤ i < n , denote by $$f_i$$ f i the number of subsets $$\sigma $$ σ of $$\{1,\ldots ,n\}$$ { 1 , … , n } of cardinality $$i+1$$ i + 1 that satisfy $$\bigcap _{i \in \sigma } A_i \ne \emptyset $$ ⋂ i ∈ σ A i ≠ ∅ . Let $$k \ge 2$$ k ≥ 2 be an integer. We prove that if each k -wise and $$(k{+}1)$$ ( k + 1 ) -wise intersection of sets from $${\mathcal {A}}$$ A is empty, or a single point, or both open and path-connected, then $$f_{k+1}=0$$ f k + 1 = 0 implies $$f_k \le cf_{k-1}$$ f k ≤ c f k - 1 for some positive constant c depending only on k . Similarly, let $$b \ge 2$$ b ≥ 2 , $$k > 2b$$ k > 2 b be integers. We prove that if each k -wise or $$(k{+}1)$$ ( k + 1 ) -wise intersection of sets from $${\mathcal {A}}$$ A has at most b path-connected components, which all are open, then $$f_{k+1}=0$$ f k + 1 = 0 implies $$f_k \le cf_{k-1}$$ f k ≤ c f k - 1 for some positive constant c depending only on b and k . These results also extend to two-dimensional compact surfaces.

中文翻译：

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平面集的交集模式

令 $${\mathcal {A}}=\{A_1,\ldots ,A_n\}$$ A = { A 1 , … , A n } 是平面中的一组集合。对于 $$0 \le i < n$$ 0 ≤ i < n ，用 $$f_i$$ fi 表示 $$\{1,\ldots ,n\}$$ 的子集数 $$\sigma $$ σ { 1 , … , n } 满足 $$\bigcap _{i \in \sigma } A_i \ne \emptyset $$ ⋂ i ∈ σ A i ≠ ∅ 的基数 $$i+1$$ i + 1 。令 $$k \ge 2$$ k ≥ 2 为整数。我们证明，如果来自 $${\mathcal {A}}$$ A 的集合的每个 k -wise 和 $$(k{+}1)$$ ( k + 1 ) -wise 交集是空的，或者一个点，或既开放又连接路径，则 $$f_{k+1}=0$$ fk + 1 = 0 意味着 $$f_k \le cf_{k-1}$$ fk ≤ cfk - 1 对于某些正常数c 仅取决于 k 。类似地，让 $$b \ge 2$$ b ≥ 2 , $$k > 2b$$ k > 2 b 是整数。我们证明如果来自 $${\mathcal {A}}$$ A 的集合的每个 k -wise 或 $$(k{+}1)$$ ( k + 1 ) -wise 交集至多有 b 个路径连接组件，它们都是开放的，那么 $$f_{k+1}=0$$ fk + 1 = 0 意味着 $$f_k \le cf_{k-1}$$ fk ≤ cfk - 1 对于某些正常数 c 取决于仅在 b 和 k 上。这些结果也扩展到二维紧凑表面。