Fooling sets and rank

An $n\times n$ matrix $M$ is called a \textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\ell} M_{\ell,k} = 0$ for every $k\ne \ell$. Dietzfelbinger, Hromkovi{\v{c}}, and Schnitger (1996) showed that $n \le (\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $\mbox{rk} M$ can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = \binom{\mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of matrices with $n= (1+o(1))(\mbox{rk} M)^2$.

💡 Research Summary

The paper investigates the fundamental relationship between the size of a “fooling‑set matrix” and its rank over an arbitrary field. A square matrix (M) of order (n) is called a fooling‑set matrix if (i) every diagonal entry (M_{i,i}) is non‑zero and (ii) for any distinct indices (k\neq\ell) the product (M_{k,\ell}M_{\ell,k}=0). This definition captures a non‑symmetric zero‑product condition that appears in many lower‑bound arguments in communication complexity, circuit complexity, and combinatorial optimization.

In 1996, Dietzfelbinger, Hromkovič, and Schnitger proved the universal inequality
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