An intermediate case of exponential multivalued forbidden matrix configuration

The forbidden number forb$(m,F)$, which denotes the maximum number of distinct columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\cdots,r-1\}$. forb$(m,r,F)$ is the maximum number of distinct columns in an $r$-matrix with no submatrix that is a row and column permutation of $F$. While forb$(m,F)$ is polynomial in $m$, forb$(m,r,F)$ is exponential for $r\geq 3$. Recently, forb$(m,r,F)$ was studied for some small $(0,1)$-matrices $F$, and exact values were determined in some cases. In this paper we study forb$(m,r,M)$ for $M=\begin{bmatrix}0&1\\0&1\\1&0\end{bmatrix}$, which is the smallest matrix for which this forbidden number is unknown. Interestingly, it turns out that this problem is closely linked with the following optimisation problem. For each triangle in the complete graph $K_m$, pick one of its edges. Let $m_e$ denote the number of times edge $e$ is picked. For each $\alpha\in\mathbb{R}$, what is $H(m,\alpha)=\max\sum_{e\in E(K_m)}\alpha^{m_e}$? We establish a relationship between forb$(m,r,M)$ and $H(m,(r-1)/(r-2))$, find upper and lower bounds for $H(m,\alpha)$, and use them to significantly improve known bounds for forb$(m,r,M)$.