Christopher Bouchard

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math.CO2026

Supersaturation in union-closed families of sets

Christopher Bouchard

Let $k$ and $n$ be positive integers such that $2 \leq k \leq n + 1$ . We prove that the number of $k$ -chains in a union-closed family with universe $[n]$ and size $m$ is minimized wh…

math.CO2025

An upper bound for union-closed family size

Christopher Bouchard

Let $A$ be a union-closed family of sets with universe $⋃_{A \in A} A = [n] = {1, \dots, n}$ and length $ℓ$ . We prove that $|\mathcal{A}| \leq \sum_{i=…

math.CO2025

An averaging result for union-closed families of sets

Christopher Bouchard

Let $A$ be a union-closed family of sets with base set $b (A) = ⋃_{A \in A} A$ denoted by $[n] = {1, \dots, n}$ , and for any real $x > 0$ , let $\math…

math.CO2025

On the lattice formulation of the union-closed sets conjecture

Christopher Bouchard

The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every fin…