We solve a generalised form of a conjecture of Kalai motivated by attempts to improve the bounds for Borsuk's problem. The conjecture can be roughly understood as asking for an analogue of the Frankl-Rodl forbidden intersection theorem in which set intersections are vector-valued. We discover that the vector world is richer in surprising ways: in particular, Kalai's conjecture is false, but we prove a corrected statement that is essentially best possible, and applies to a considerably more general setting. Our methods include the use of maximum entropy measures, VC-dimension, Dependent Random Choice and a new correlation inequality for product measures.

中文翻译：

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禁止向量值相交

我们通过尝试改善Borsuk问题的范围来解决Kalai猜想的一般形式。该猜想可以粗略地理解为要求一个Frankl-Rodl禁止的交集定理的类似物，在该定理中，集合交集是矢量值的。我们发现向量世界以令人惊讶的方式变得更加丰富：特别是，Kalai的猜想是错误的，但是我们证明了一种更正确的陈述，该陈述实质上是最有可能的，并且适用于相当普遍的环境。我们的方法包括使用最大熵测度，VC维，相关随机选择和乘积测度的新相关不等式。