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On the arithmetic mean of the size of cross-union families
Acta Mathematica Hungarica ( IF 0.6 ) Pub Date : 2021-05-05 , DOI: 10.1007/s10474-021-01138-6
P. Frankl

Let \(n > k > 1\) be integers, \([n] = \{1, \ldots, n\}\) the standard \(n\)-element set and \({[n]\choose k}\) the collection of all its \(k\)-subsets. The families \(\mathcal F_0, \ldots, \mathcal F_s \subset {[n]\choose k}\) are said to be cross-union if \(F_0 \cup \cdots \cup F_s \neq [n]\) for all choices of \(F_i \in \mathcal F_i\). It is known [13] that for \(n \leq k(s + 1)\) the geometric mean of \(|\mathcal F_i|\) is at most \({n - 1\choose k}\). We conjecture that the same is true for the arithmetic mean for the range \(ks < n < k(s + 1)\), \(s > s_0(k)\) (Conjecture 8.1) and prove this in several cases. The proof for the case \(n = ks + 2\) relies on a novel approach, a combination of shifting and Katona’s cyclic permutation method.



中文翻译:

关于跨联盟家庭规模的算术平均值

\(n> k> 1 \)为整数,\([n] = \ {1,\ ldots,n \} \)为标准\(n \)元素集,而\({[n] \ choose k} \)所有\(k \)-子集的集合。家庭\(\ mathcal F_0,\ ldots,\ mathcal F_S \子集{[N] \选择k} \)被说成是交联合如果\(F_0 \杯\ cdots \杯F_S \ NEQ [N] \ )\(F_i \ in \ mathcal F_i \)中的所有选择。已知[13]对于\(n \ leq k(s + 1)\)\(| \ mathcal F_i | \)的几何平均值最多为 \({n -1 \ choose k} \)。我们猜想范围的算术平均值也是如此 \(ks <n <k(s + 1)\)\(s> s_0(k)\)(猜想8.1)并在几种情况下证明这一点。\(n = ks + 2 \)的情况的证明依赖于一种新颖的方法,即位移法和卡托纳循环置换法的结合。

更新日期:2021-05-06
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