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Theorem 1.

Convex optimisation bound If $r,l \in \mathbb{N}$ maximise $mr + l$ subject to $r \le n$ and $l < m$, and $$(m-l){r \choose t} + l{r+1 \choose t} \le (s-1){n \choose t}$$ then $z(m,n;~s,t) \le mr + l$.

Let $\mathbf{r} \in \mathbb{N}^m$ be the vector of row weights of an $(s,t)$-rectangle-free matrix. We have$$\sum_{i=1}^m {r_i \choose t} \le (s-1){n \choose t}$$otherwise some combination of $t$ columns forms a rectangle with some $s$ rows by the pigeonhole principle. As this sum is a convex function of $\mathbf{r}$, without loss of generality $\sum r_i$ is maximised when every $\left|r_i - r_j\right| \le 1$. In this case we have $l$ rows of some weight $r+1$, and the remaining $m-l$ rows of weight $r$.

Theorem 2.

Čulik's theorem [Cul56] If $m \ge (s-1){n \choose t}$ then $z(m,n;~s,t) = (t-1)m + (s-1){n \choose t}$.

The upper bound given by Theorem 1 has $r = t-1$ and $l = (s-1){n \choose t}$, which is attainable by using each combination of $t$ columns in $s-1$ rows.

Theorem 3.

Reiman's theorem [Rei58] For $k \ge 2$, $$z(k^2 + k + 1,~k^2 + k + 1) \le (k+1)(k^2 + k + 1)$$and equality is acheived by and only by the incidence matrix of a finite projective plane of order $k$.

This is the upper bound given by Theorem 1, and is only achieved by a vector $\mathbf{r}$ with every $r_i=k+1$. As $(k^2 + k+1){k+1 \choose 2} = {k^2 + k + 1 \choose 2}$, every pair of columns must intersect in some row, as required for a finite projective plane.


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