North-East Lattice Paths with Few Collinear Vertices

Both bounds move, and the gap stays enormous: the lower bound rises from $\exp(\Omega(\log^2 k))$ to $\exp(\Omega(k^{1/3}))$ and the upper falls from $\exp(O(k^4))$ to $\exp(O(k^2))$, so $A(k)$ is still undetermined between an exponent of $k^{1/3}$ and one of $k^2$. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness, a sharper line-counting step, or a different model entirely.

A named problem - the Gerver-Ramsey collinearity problem - from two 1979 Pacific J. Math. papers, catalogued in Brass-Moser-Pach's standard problem book and still drawing work in 2024 and 2026. Forty-seven years open with a genuine literature, but firmly inside discrete geometry: the named specialist band at 15.