# MathCS Seminar

Title: On the Number of B_h Sets of a Given Size
Seminar: Combinatorics
Speaker: Domingos Dellamonica of Sao Paulo
Contact: Vojtech Rodl, rodl@mathcs.emory.edu
Date: 2014-10-27 at 4:00PM
Venue: W303
Abstract:
For an integer h bigger or equal to 2, a \$B_h\$ set is a set of integers with the property that every collection containing h of its elements yield a unique sum (and repetitions are allowed). For \$h = 2\$, such sets are also called Sidon sets. In this talk we will describe our recent results on estimating \$F(n, s, h)\$, which we define as the number of \$B_h\$ sets of cardinality s containing integers from \$[n] = {1, 2, ..., n}\$. It is not hard to see that for \$s > n^(1/h)\$, we have \$F(n, s, h) = 0\$. Indeed, in this case there are more h-sums than possible outcomes for the sums. On the other hand, there are constructions of B-h sets having cardinality \$c.n^(1/h)\$, (with c depending on h only) hence we shall estimate the behavior of \$F(n, s, h)\$ for s up to \$O( n^(1/h))\$. Our counting shows the existence of a surprising threshold function \$T(n)\$: for values of \$s << T(n)\$, the B-h sets are abundant while for \$s >> T(n)\$ the B-h sets become very rare. More precisely, we show that \$T(n) ~ n^{(1 + o(1))/(2h - 1)}\$ and establish fairly precise estimates of \$F(n, s, h)\$ for the entire range of s.