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Infinite Sidon Sets Contained in Sparse Random Sets of Integers
Kohayakawa, Yoshiharu,Lee, Sang June,Moreira, Carlos Gustavo,Rö,dl, Vojtě,ch Society for Industrial and Applied Mathematics 2018 SIAM Journal on Discrete Mathematics Vol.32 No.1
<P>A set <TEX>$S$</TEX> of natural numbers is a <italic toggle='yes'>Sidon</I> set if all the sums<TEX>$s_1+s_2$</TEX> with <TEX>$s_1$</TEX>, <TEX>$s_2\in S$</TEX> and <TEX>$s_1\leq s_2$</TEX> are distinct.Let constants <TEX>$\alpha>0$</TEX> and <TEX>$0<\delta<1$</TEX> be fixed, andlet <TEX>$p_m=\min\{1,\alpha m^{-1+\delta}\}$</TEX> for all positiveintegers <TEX>$m$</TEX>. Generate a random set <TEX>$R\subset {\mathbb N}$</TEX> by adding <TEX>$m$</TEX>to <TEX>$R$</TEX> with probability <TEX>$p_m$</TEX>, independently for each <TEX>$m$</TEX>. Weinvestigate how dense a Sidon set <TEX>$S$</TEX> contained in <TEX>$R$</TEX> can be. Ourresults show that the answer is qualitatively very different in atleast three ranges of <TEX>$\delta$</TEX>. We prove quite accurate results forthe range <TEX>$0<\delta\leq2/3$</TEX>, but only obtain partial results forthe range <TEX>$2/3<\delta\leq1$</TEX>.</P>
The number of B<sub>3</sub>-sets of a given cardinality
Dellamonica, D.,Kohayakawa, Y.,Lee, S.J.,Rodl, V.,Samotij, W. Academic Press 2016 Journal of combinatorial theory. Series A Vol.142 No.-
<P>A set S of integers is a B-3-set if all the sums of the form a(1) a(2)+a(3), with a(1), a(2) and a(3) epsilon S and a(1) <= a(2) <= a(3), are distinct. We obtain asymptotic bounds for the number of B-3-sets of a given cardinality contained in the interval [n] = {1,...,n}. We use these results to estimate the maximum size of a B-3-set contained in a typical (random) subset of [n] of a given cardinality. These results confirm conjectures recently put forward by the authors [On the number of B-h-sets, Combin. Probab. Comput. 25 (2016), no. 1, 108-127]. (C) 2016 Elsevier Inc. All rights reserved.</P>
On the Number of <i>B<sub>h</sub></i>-Sets
DELLAMONICA Jr, DOMINGOS,KOHAYAKAWA, YOSHIHARU,LEE, SANG JUNE,RÖ,DL, VOJTĚ,CH,SAMOTIJ, WOJCIECH Cambridge University Press 2016 Combinatorics, probability & computing Vol.25 No.1
<P>A set <I>A</I> of positive integers is a <I>Bh-set</I> if all the sums of the form <I>a</I>1 + . . . + <I>ah</I>, with <I>a</I>1,. . .,<I>ah</I> ∈ <I>A</I> and <I>a</I>1 ⩽ . . . ⩽ <I>ah</I>, are distinct. We provide asymptotic bounds for the number of <I>Bh</I>-sets of a given cardinality contained in the interval [<I>n</I>] = {1,. . .,<I>n</I>}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of <I>Bh</I>-sets. We also use these results to estimate the maximum size of a <I>Bh</I>-sets contained in a typical (random) subset of [<I>n</I>] with a given cardinality.</P>