The Uniformity Conjecture over the rationals

Let $g\geq 2$ be an integer. A smooth curve is a curve defined over $\mathbb{Q}$, and its rational points are its points over $\mathbb{Q}$. Uniformity Conjecture over $\mathbb{Q}$. For every integer $g\geq 2$ there exists an integer $B_g$ such that any smooth curve defined over $\mathbb{Q}$ has at most $B_g$ rational points.

Faltings's theorem gives finiteness of the rational points on each smooth curve of genus at least $2$, but the bound may depend on the curve. The conjecture asserts uniformity in the curve; it would follow from the Bombieri–Lang conjecture and is used in the paper for conditional combinatorial results.