Open-problem statements, with their sources attached.

Erdős Problem 872, part (i) (weak form): there exists a constant $\epsilon > 0$ such that the game length is at least $\epsilon \cdot n$ for all sufficiently large $n$.

Erdős Problem 872, part (ii) (strong form): for every $\epsilon > 0$, the game length is at least $(1-\epsilon) \cdot n / 2$ for all sufficiently large $n$.

Status note: the forum thread (April-May 2026) records Shortener strategies giving $L(n) \leq (23/48 + o(1)) \cdot n$ (described in the thread as accepted as correct, with a Lean formalization in progress) and a claimed $L(n) \leq 0.19 \cdot n$, either of which would answer this question negatively under the Prolonger-first convention. Neither is published, so the statement is recorded here as the original Erdős question.

Forum-related variant: how small can a maximal primitive subset of $\{2, \dots, n\}$ be? The set of primes in $\{2, \dots, n\}$ is a maximal primitive subset of size $\pi(n)$, and the forum thread asks whether this is the smallest possible for all $n \geq 2$. Equivalently: must every completed play of the saturation game, by both players and regardless of strategy, claim at least $\pi(n)$ elements? (Terminal positions of the game are exactly the maximal primitive subsets.)

Let $T_N = \sum_{k=1}^{N} k = \frac{N(N+1)}{2}$. If $T_N$ is even (equivalently $N \equiv 0 \pmod 4$ or $N \equiv 3 \pmod 4$), then under optimal play the game Catch-Up($\{1, \ldots, N\}$) ends in a draw.