A sunflower with p petals consists of p sets whose pairwise intersections are identical. The goal of the sunflower problem is to find the smallest r=r(p,k) such that any family of r^k distinct k-element sets contains a sunflower with p petals. Building upon a breakthrough of Alweiss, Lovett, Wu and Zhang from 2019, Rao proved that r=O(p log(pk)) suffices; this bound was reproved by Tao in 2020. In this short note we record that r=O(p log k) suffices, by using a minor variant of the probabilistic part of these recent proofs.
翻译：带有花瓣的向日葵由p组组成, 其相近交叉点是相同的。 向日葵问题的目标是找到最小的 r=r( p, k), 这样任何 rk 不同的 k 元素组的家族都包含一个带有p spetal 的向日葵。 在Alweis、 Lovett、 Wu 和 Zhang 从2019年突破后, Rao 证明r=O( p log( pk) ) 已经足够; 2020年, Tao 重新确认了这一约束。 在这个简短的注释中,我们记录到r=O( p log k) 足够, 使用这些最近证据的概率部分的微小变量。
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