Abstract
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.
Highlights
Intersection theorems have been a central topic of Extremal Combinatorics since the seminal paper of Erdos, Ko and Rado [9], and the area has grown into a vast body of research
as s ∈S μps (As) key paradigm of our approach is that V-intersection theorems often have equivalent formulations in terms of certain product measures, and that the necessary condition for these theorems appears naturally as a condition on the product measures. (A similar idea arose in the new proof of the density Hales–Jewett theorem developed by the first Polymath project [26], in this case the natural ‘equal slices’ distribution was not a product measure.)
There are several natural directions in which potential generalisations of our results can be explored: instead of associating vectors in ZD to each coordinate we may consider values in another group G, and we may consider more general functions of the coordinate values, for example, a polynomial rather than a linear function. (Is there a ‘local’ version of Kim-Vu [24] polynomial concentration?) Even for linear functions in one dimension, our setting seems somewhat related to some open problems in additive combinatorics, such as the independence number of Paley graphs, but here our assumptions seem too restrictive
Summary
Intersection theorems have been a central topic of Extremal Combinatorics since the seminal paper of Erdos, Ko and Rado [9], and the area has grown into a vast body of research (see [2, 4, 19] for an overview). In a recent survey on the Borsuk problem, Kalai [21] remarked that the Frankl–Rodl theorem can be used to give a counterexample to the Borsuk conjecture (the Frankl–Wilson intersection theorem [13] was used in Kahn and Kalai’s celebrated counterexample [20]), and suggested that improved bounds might follow from a suitably generalised Frankl–Rodl theorem. While Theorem 1.3 shows that Conjecture 1.2 is generally false, for expository purposes we will give two concrete counterexamples (one of which draws a straightforward analogy with the Frankl–Rodl setting), illustrating two different reasons why the conjecture fails.
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