A family of vectors in [k]n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k]n invariant under a transitive group of symmetries is o(kn), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.

中文翻译：

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关于对称相交向量族

[ 中的向量族ķ]n据说是相交如果它的任何两个元素在至少一个坐标上一致。我们证明，对于固定ķ≥ 3，即 [ķ]n在传递对称群下的不变量是○(ķn)，这与布尔超立方体的情况（其中ķ= 2)。我们的主要贡献解决了现有技术的局限性：虽然现在有一些方法，首先出现在 Ellis 和第三作者的工作中，用于使用谱机制来解决涉及对称性的极值集理论中的问题，但这种机制主要依赖于 up-集合、有偏差的乘积度量和布尔超立方体中的阈值行为，这些特征在此处考虑的问题中明显不存在。为了绕过这些障碍，引入似乎具有独立兴趣的想法，我们开发了一种适用于 posets 产品级别的尖锐阈值机制的变体。