论文标题
弗兰克对跨工会家庭的猜想的证明
A proof of Frankl's conjecture on cross-union families
论文作者
论文摘要
The families $\mathcal F_0,\ldots,\mathcal F_s$ of $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ are called cross-union if there is no choice of $F_0\in \mathcal F_0, \ldots, F_s\in \mathcal F_s$ such that $ f_0 \ cup \ ldots \ cup f_s = [n] $。著名的Erdős-Ko-Rado定理的自然概括，由于Frankl和Tokushige，指出，对于$ n \ le（s+1）k $，几何平均值$ \ lvert \ mathcal f_i \ rvert $最多是$ \ binom {n-b-binom {n-1} {k} {k} $。弗兰克（Frankl）猜想，在某些轻度条件下，算术平均值应该相同。我们通过证明跨加基族家族的算术平均值的唯一（达到同构）最大化是自然的一个$ \ MATHCAL F_0 = \ ldots = \ Mathcal f_s = {[N-1] \ select K} $，这证明了弗兰克的猜想。
The families $\mathcal F_0,\ldots,\mathcal F_s$ of $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ are called cross-union if there is no choice of $F_0\in \mathcal F_0, \ldots, F_s\in \mathcal F_s$ such that $F_0\cup\ldots\cup F_s=[n]$. A natural generalization of the celebrated Erdős--Ko--Rado theorem, due to Frankl and Tokushige, states that for $n\le (s+1)k$ the geometric mean of $\lvert \mathcal F_i\rvert$ is at most $\binom{n-1}{k}$. Frankl conjectured that the same should hold for the arithmetic mean under some mild conditions. We prove Frankl's conjecture in a strong form by showing that the unique (up to isomorphism) maximizer for the arithmetic mean of cross-union families is the natural one $\mathcal F_0=\ldots=\mathcal F_s={[n-1]\choose k}$.
