Intersection theorems with geometric consequences

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“…Our method of bounding the rank of the matrices, in spirit, resembles the techniques in the papers [7,8] by Frankl and Wilson. In these papers, the authors consider the problem of bounding the sizes of some interesting families of sets. They reduce this problem to analyzing ranks of certain classes of matrices and use the linear algebra method [3] to bound the ranks.…”

Ослабленный Закон Нуля Или Единицы Для Последовательностей Случайных Дистанционных Графов

“…С помощью этого графа они показали, что хроматическое число пространства R n растет экспоненциально (см. [15]). В 1991 г. Дж.…”

“…В силу того что n делится на 8, равенства возможны тогда и только тогда, когда n ≡ 16 (mod 24). Элементарный перебор остальных вариантов (x 3 3, x 9 4, x 10 4, x 15 4, x 16 4) показывает, что в рассматриваемом случае всегда можно подобрать нужный вектор y.…”

Explicit Constructions

“…This is the problem of obtaining constructive lower bounds for the usual diagonal Ramsey numbers. Specifically, we want to describe explicitly, for every k, a graph with c k nodes that contains neither a clique of size k, nor a stable set of size k, where c > 1 is a constant, independent of k. The best known result in this direction is that of Frankl and Wilson (1981), who constructed such a graph with exp Ω(log 2 k/ log log k) nodes. It may be true that for primes q, the Paley graphs G q are better examples, but, at present, a proof of this, which would have several new number-theoretic consequences, seems hopeless.…”