North-East Lattice Paths with Few Collinear Vertices

Let $A(k)$ be the largest possible number of moves in a north-east lattice path whose visited vertices contain no $k$ collinear points. Gerver (1979) and Gerver and Ramsey (1979) gave lower and upper bounds on $A(k)$ of the form $$\exp\left(\Omega(\log(k)^2)\right)\le A(k)\le \exp\left(O(k^4)\right).$$ Improving upon these results, we show that $$\exp\left(\Omega(k^{1/3})\right)\le A(k)\le \exp\left(\left(\frac{2}{e}+o(1)\right)(k-1)^2\right).$$