A note on Erdős matrices and Marcus–Ree inequality

In 1959, Marcus and Ree proved that any bistochastic matrix A satisfies${\Delta }_n\left(A\right)≔\underset{\sigma \in S_n}{\max }\sum _{i=1}^{n}A(i,\sigma (i\left)\right)-\sum _{i,j=1}^{n}A{(i,j)}^2\ge 0.$ Erdős asked to characterize the bistochastic matrices satisfying ${\Delta }_n\left(A\right)=0$. This problem remains largely open, and very recently, a complete list of such matrices was obtained in dimension $n=3$ by Bouthat, Mashreghi, and Morneau-Guérin. Soon after, Tripathi proved that there were only finitely many such matrices in any dimension n. In this paper, we continue the investigation initiated in these two works. We characterize all $4\times 4$ bistochastic matrices satisfying ${\Delta }_4\left(A\right)=0$. Furthermore, we show that for $n\ge 3$, ${\Delta }_n\left(A\right)=\alpha$ has uncountably many solutions when $\alpha \in (0,(n-1)/4)$. This answers a question raised in (Tripathi, 2025 [16]). We also extend the Marcus–Ree inequality to infinite bistochastic arrays and bistochastic kernels. Our investigation into $4\times 4$ Erdős matrices also leads to several intriguing questions of independent interest. We propose several questions and conjectures and present numerical evidence for them.