Take a 1 by 1 box $D \subset \mathbb{R}^2$ and let $U_1,\dots,U_n$ be i.i.d. uniforms in $D$.

Suppose at the start all of $\mathcal{V}_0=\{U_1,\dots,U_n\}$ are viable. At each step pick one of the viable uniform random variables $U=(U^1,U^2)$ according to some strategy. Say this point is eaten. The viable points are now $\mathcal{V}_1:=\{V=(V^1,V^2) \in \mathcal{V}: V^1\geq U^1 \text{ and } V^2 \geq U^2\}$. Carry on in this manner until there no more viable points.

Let $M(n)$ denote the maximum number of eaten points over all strategies. This is the maximal number of points visited on a directed crossing from one corner of the box to the opposite corner.

I am interested in the behavior of $\mathbb{E}[M(n)]$ asymptotically in $n$.

I have a picture of this below. Here, there are many strategies that maximise the number of eaten points, which is 3, one of which is drawn in red.

I expect that $\mathbb{E}[M(n)] \approx C\log n$ as it seems that for an optimal strategy the proportion of points $\frac{|\mathcal{V}_{i+1}|}{|\mathcal{V}_i|}$ lost by eating a point remains roughly the same. I am having trouble showing this rigorously.