For each , form the multilinearization on the Boolean cube ofAt the characteristic vector of a set , this polynomial vanishes for and is nonzero for . Hence the restricted functions are linearly independent. On the -slice, every square-free monomial of degree below can be raised to degree using the relation , so the degree-at-most- function space is spanned by the square-free degree- monomials. Linear independence gives the theorem.
The answer is . All four-sets containing one fixed pair form a family of sizewhose distinct intersections have size two or three. The Ray-Chaudhuri–Wilson theorem for the two allowed intersection sizes gives the matching upper bound.
The answer is . The familyhas size and pairwise intersection three. For the upper bound, work modulo two. Every member has size modulo two, while every allowed intersection has residue . The Frankl-Wilson theorem with gives .
The answer is . Partition all but at most one point into disjoint pairs and take all unions of two pairs. There are such four-sets, and two distinct unions intersect in zero or two points. The Ray-Chaudhuri–Wilson theorem gives the upper bound .
The answer is . Take a Steiner triple system, or a partial one of quadratic size, on and adjoin the fixed point to every triple. Distinct triples meet in zero or one point, so the resulting four-sets meet in one or two points. This gives members. The Ray-Chaudhuri–Wilson theorem with the two allowed intersection sizes gives .
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