Sharp Bounds for Sets with Distinct Subset Products

TL;DR

This work resolves a question of Erdős by determining the asymptotic size of the largest subset $A\subseteq[N]$ with distinct subset products. The authors introduce a prime factorization graph $G(A)$ that encodes the large and medium prime factors of elements of $A$ and exploit the constraint of distinct subset products to force a scarcity of short cycles in $G(A)$. Through a cycle-removal procedure and a careful edge-count bound for graphs without short cycles, they establish $|A|\le \pi(N)+\tfrac{1}{2}\pi(N^{1/2})+\tfrac{1}{2}|\mathcal{P}_{\square}|+O(N^{5/12+o(1)})$, yielding the main result $f(N)=\pi(N)+\pi(N^{1/2})+O(N^{5/12+o(1)})$ (and the squarefree refinement when medium-prime squares are forbidden). The paper also provides constructions showing near-sharpness and expands the understanding of lower-order terms via new examples, including a $\tfrac{1}{3}\pi(N^{1/3})$-type contribution. Overall, the graph-theoretic framework yields sharp upper bounds and near-matching lower bounds for this extremal subset-product problem.

Abstract

Let $A\subseteq [N]$ be such that for any pair of distinct subsets $B,C\subset A$, the products $\prod_{b\in B}b$ and $\prod_{c\in C}c$ are distinct. We prove that $|A|\leq π(N)+π(N^{1/2})+o(π(N^{1/2}))$, where $π$ is the prime counting function, answering a question of Erdős.

Theorems & Definitions (38)

• Example 1.1
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 1.7: Subset Product Set
• Definition 1.8: Small, Medium, and Large Primes, Valuations
• Definition 1.9: Big-O and Little-o Notation
• Definition 1.10: Paths, Cycles, Circuits
• Proposition 2.1