A note on Erdős matrices and Marcus\unicode{x2013}Ree inequality
In 1959, Marcus and Ree proved that any bistochastic matrix $A$ satisfies $Δ_n(A):= \max_{σ\in S_n}\sum_{i=1}^{n}A(i, σ(i))-\sum_{i, j=1}^n A(i, j)^2 \geq 0$. Erdős asked to characterize the bistochastic matrices satisfying $Δ_n(A)=0$. 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-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 $Δ_4(A)=0$. Furthermore, we show that for $n\geq 3$, $Δ_n(A)=α$ has uncountably many solutions when $α\in (0, (n-1)/4)$. This answers a question raised in [Tripathi, R., $\textit{Some observations on Erdős matrices}$, 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 $4\times 4$ Erdős matrices also leads to several intriguing questions of independent interest. We propose several questions and conjectures and present numerical evidence for them.