Result: The Erdős--Selfridge Bound for Product-Free Sets and Multiplicative Sidon Sets: An Elementary Approach

This is the preprint location for the manuscript submitted to the Integers: Electronic Journal of Combinatorial Number Theory. This preprint gives an elementary proof that any product-free subset of the natural numbers has upper density at most 1/2. The argument uses a contradiction based on elementary divisor estimates, with a fully self-contained derivation of the key bounds. The paper also constructs a product-free set with density exactly 1/2, showing the bound is sharp, and proves that multiplicative Sidon sets have zero upper density. All proofs are elementary and accessible to readers in number theory and combinatorics.