A note on Erd\H{o}s matrices and Marcus\unicode{x2013}Ree inequality

In 1959, Marcus and Ree proved that any bistochastic matrix $A$ satisfies $\Delta_n(A):= \max_{\sigma\in S_n}\sum_{i=1}^{n}A(i, \sigma(i))-\sum_{i, j=1}^n A(i, j)^2 \geq 0$. Erd\H{o}s asked to characterize the bistochastic matrices satisfying $\Delta_n(A)=0$. 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-Gu\'erin. 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 $\Delta_4(A)=0$. Furthermore, we show that for $n\geq 3$, $\Delta_n(A)=\alpha$ has uncountably many solutions when $\alpha\in (0, (n-1)/4)$. This answers a question raised in [Tripathi, R., $\textit{Some observations on Erd\H{o}s matrices}$, Linear Algebra and Its Applications 708 (2025)]. We also extend the Marcus\unicode{x2013}Ree inequality to infinite bistochastic arrays and bistochastic kernels. Our investigation into $4\times 4$ Erd\H{o}s matrices also leads to several intriguing questions of independent interest. We propose several questions and conjectures and present numerical evidence for them.