Probabilistic results Sample Clauses

Probabilistic results. We investigate Sidon subsets of sparse, random sets of integers, that is, we replace the ‘environment’ [n] by a sparse, random subset R of [n], and ask how large a subset S ⊂ R can be, if we require that S should be a Sidon set. Let us now state a weak, but less technical version of our main probabilistic results. Let F (R) = max |S|, where the maximum is taken over all Sidon subsets S ⊂ R. Let [n]m be a random subset of [n] of cardinality m = m(n), with all the subsets of [n] equiprobable. We are interested in the random variable F ([n]m). The usual deletion method gives that, almost surely, that is, with probability tending to 1 as n → ∞, we have F ([n]m) = (1−o(1))m if m = m(n) n1/3. On the other hand, the results of Xxxxxxx [40] and Xxxxxx and Gowers [12] imply that, if m = m(n) n1/3, then, almost surely, we have
Sample 1
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Probabilistic results. We investigate B3-sets contained in a sparse, random set of [n], that is, we replace the ‘environment’ [n] by a sparse, random subset [n]m of [n], where [n]m denotes a random subset of [n] of cardinality m = m(n), with all the subsets of [n] with size m having the same probability. Then we ask how large a subset S ⊂ [n]m can be, if we require that S should be a B3-set. The following definition will provide a notation suitable for this problem.
Sample 1

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