Combinatorics Seminar: Two problems regarding primitive sets of integers

Title: Two problems regarding primitive sets of integers
Speaker: Nathan McNew, Towson
Date and time: Monday, April 15, 4–5pm
Place: Phillips 736

Abstract: A set of integers is primitive if no integer in the set divides another. The prime numbers form one such a set, but there are many others. Let f(n) count the number of primitive subsets of the integers {1, 2, ... n}. Cameron and Erdos showed that 1.55^n < f(n) < 1.59^n and conjectured that f(n) was asymptotic to c^n for some constant c. In 2017 the conjecture was proven Angelo, but his proof is ineffective in the sense that it gives no information about the value of the constant c. We improve Angelo's result by showing how this constant can be computed and obtaining an error term. Then, we use the probabilistic method to construct infinite primitive sets with relatively small gaps between consecutive terms, substantially smaller than is known to hold for the primes. The techniques we present generalize to a variety of related combinatorial questions.