# Monochromatic Edges in Complete Multipartite Hypergraphs
Consequently, it is only possible when \(n_{1}=3\). Note that \(n_{2}<n_{3}\) and \(n_{1}\ +\ n_{2}=3+n_{2}>n_{3}\). For \(N=3+n_{2}+n_{3}\) to be odd, \(n_{2}\) and \(n_{3}\) must have the same parity. Thus, \(n_{3}=n_{2}+2\) and \(5>n_{1}+n_{3}-n_{2}=3+n_{2}+2-n_{2}=5\), which is a contradiction. Now, we have that \(n_{1}+n_{3}-X\geqslant 3\). Hence, \(F_{8}\) gives fewer monochromatic edges than \(F_{6}\), and \(c_{X}^{*}\) is in the form \(F_{8}\). To sum up, we have already determined (as shown in Tables 9 and 10) the canonical form \(c_{X}^{*}\) that has the minimum number of monochromatic edges for each condition of hypergraphs and range of the number of red vertices \(X\). Note that, in Case 3, \(c_{X}^{*}\) is only in the form \(F_{2}\) when \(\frac{n_{1}+n_{2}}{2}<X<n_{2}\). However, if \(X=\frac{n_{1}+n_{2}}{2}\), then \(F_{2}\) and \(F_{4}\) give the same number of monochromatic edges. Moreover, in Case 4B, we have not compared the colorings in the form \(F_{3}\) and \(F_{6}\). The uniqueness of \(c_{X}^{*}\) of the other cases will be also considered. We can see that the inequalities in some cases are equal when the sizes of some parts are equal, which means that those canonical forms are equivalent. For example, in Case 1, if \(n_{2}=n_{3}\), \(F_{2}\) is equivalent to \(F_{3}\). The remaining \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cases** & **Number of Red Vertices** & **Canonical Form** \\ \hline
1 & \(0\leqslant X<n_{1}\) & \(F_{1}\) \\
2 & \(n_{1}\leqslant X<\frac{n_{1}+n_{2}}{2}\) & \(F_{4}\) \\
3 & \(X=\frac{n_{1}+n_{2}}{2}\) & \(F_{2}\) or \(F_{4}\) \\  & \(\frac{n_{1}+n_{2}}{2}<X<n_{2}\) & \(F_{2}\) \\
4 & \(n_{2}\leqslant X<n_{1}+n_{2}\) & \(F_{6}\) \\
5 & \(n_{1}\ +\ n_{2}\leqslant X\leqslant\lfloor\frac{N}{2}\rfloor\) & \(F_{10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: List of best canonical forms for Type A hypergraphs \(n_{1}+n_{2}\leqslant n_{3}\). inequalities are equal when \(X\) is equal to some certain value, such as in Case 5A and Case 5B when \(X=\frac{N}{2}\); the colorings in the forms \(F_{3}\) and \(F_{10}\) are isomorphic up to a permutation of the name of colors and so are colorings in the forms \(F_{6}\) and \(F_{8}\) in Case 5B. Hence, apart from Case 3, where \(X=\frac{n_{1}+n_{2}}{2}\), and Case 4B, a coloring with red vertices in the form, according to the previous table, has fewer monochromatic edges than other colorings with the same amount of red vertices and the uniqueness follows. Next, we will make comparisons between colorings with different values of \(X\). We will show that any \(c_{X}^{*}\) with \(X\leqslant\lfloor\frac{N}{2}\rfloor-1\) has strictly more monochromatic edges than some colorings. We, hence, would like to show that \(M(H,c_{X+1}^{*})-M(H,c_{X}^{*})<0\) for each \(0\leqslant X<\lfloor\frac{N}{2}\rfloor-1\). Note that, if \(c_{X}^{*}\) and \(c_{X+1}^{*}\) are in the same canonical form, we will consider \(\triangle M(H,c_{X}^{*})\) instead. Again, we will divide into several cases conforming to the value of \(X\) and the type of \(H\). Case 1: \(0\leqslant X<n_{1}\). We have that \(c_{X}^{*}\) is in the form \(F_{1}\) with \(x_{1}=X\), \(x_{2}=0\) and \(x_{3}=0\). Then,  
\[\triangle_{1}M(H,c_{X}^{*})=\left[\binom{X}{2}-\binom{X}{2}\right]-\left[ \binom{N-X\ -\ 1}{2}-\binom{n_{1}-X\ -\ 1}{2}\right]\leqslant 0.\]  
The equality holds only when \(N-X\ -\ 1<2\), which is impossible. Case 2: \(n_{1}\leqslant X<\frac{n_{1}+n_{2}}{2}\). We have that \(c_{X}^{*}\) is in the form \(F_{4}\) with \(x_{1}=n_{1}\), \(x_{2}=X-n_{1}\) and \(x_{3}=0\). Then,  
\[\triangle_{2}M(H,c_{X}^{*})=\left[\binom{X}{2}+\binom{n_{1}+n_{2}-X\ -\ 1}{2} \right]-\left[\binom{X-n_{1}}{2}\right]+\ \binom{N-X\ -\ 1}{2}.\]  
We will show that \(\triangle_{2}M(H,c_{X}^{*})<0\) by Proposition 1. We have \(X\ +\ (n_{1}+n_{2}-X\ -\ 1)=n_{1}+n_{2}-1\leqslant n_{3}+n_{2}-1=(X-n_{1})+(N-X \ -\ 1)\), \(X-n_{1}<X\) and \(n_{1}+n_{2}-X\ -\ 1<N-X-1\). Since \(2\leqslant N-X\ -\ 1\), we have that \(\triangle_{2}M(H,c_{X}^{*})<0\).