On Sidon Sets in a Random Set of Vectors

The number of Bh‐sets of a given cardinality

“…Another proof of Theorem 1 in the case h = 2 was later given by Saxton and Thomason [33] (see [28] for a similar result for [n] d , d 2). Let us also mention that Saxton and Thomason [33] proved that, perhaps somewhat surprisingly, we have log 2 |Z 2 n | (1.16 + o(1))F 2 (n).…”

“…and observe that (28) implies that condition (i) from the statement of this lemma is satisfied; see the definition of ξ j in (15).…”

“…. , h − 1} and ∈ [h − 1], we have R G (k−1) ,G ( ) 1/(λ · ξ k+ ), and hence we may apply Lemma 19 with (28) such that for any…”

The number of B3-sets of a given cardinality

“…In [10,11,12], the cardinality of the largest B 2 -sets, i.e., Sidon sets, contained in random sets of integers was investigated. Given an integer function 0 ≤ m = m(n) ≤ n, let us denote by [n] m an melement subset of [n] chosen uniformly at random from all such sets.…”

“…Take q = (log n)/β = O (log n) 4 s 1−ε = o(s). 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Note that (10) is satisfied. Hence, Observation 3.2 and Lemma 3.3 yield that the number of B 3 -sets of cardinality s that contain a set T satisfying P 100λ,ε,0 with cardinality s 1−6ε is at most…”