A 7 × 7 grid with no rectangle of marked cells

Idea. Count (row, pair of marked columns in that row).

Upper bound. Let $r_i$ be the number of marked cells in row $i$, and $T=\sum r_i$. A pair of columns can appear in at most one row. There are $\binom72=21$ pairs of columns, so
$$\sum_{i=1}^7\binom{r_i}2\le21.$$
By convexity, $\sum\binom{r_i}2\ge\frac{T(T-7)}{14}$. If $T\ge22$, this is at least $\frac{22\cdot15}{14}>23>21$, a contradiction. So $T\le21$.

Construction with $21$. Number rows and columns $0,\dots,6$ and mark in row $i$ the columns $i,\ i+1,\ i+3\pmod7$. The set $\{0,1,3\}$ is a perfect difference set modulo $7$: any two rows share exactly one marked column, so there is no rectangle; $7\cdot3=21$ marked cells.

Answer. $21$. (This is the incidence matrix of the Fano plane.)

Lesson. Count pairs inside each row, apply convexity, and look for a design that achieves the bound.