It is known that the complete graph $K_{2m}$ has a 1-factorization (e.g., http://en.wikipedia.org/wiki/Graph_factorization). This means that its set of edges can be written as a disjoint union of $2m-1$ complete matchings $M_1, \dots, M_{2m-1}$, each with $m$ edges. If $e$ is an edge of $K_{2m}$ regarded as a 2-element subset $\lbrace i,j\rbrace$ of $[2m] = \lbrace 1,2,\dots, 2m\rbrace$, then let $\bar{e}$ denote the complement $[2m]-\lbrace i,j\rbrace$, and let $\bar{M}_i =\lbrace \bar{e}\colon e\in M_i\rbrace$. Color the elements of $\bar{M}_i$ with the color $i$. This give a proper coloring of the $2m-3$ simplices of $\Delta^{2m-1}$ in $2m-1$ colors.

It is known that the complete graph $K_{2m}$ has a 1-factorization (e.g., http://en.wikipedia.org/wiki/Graph_factorization). This means that its set of edges can be written as a disjoint union of $2m-1$ complete matchings $M_1, \dots, M_{2m-1}$, each with $m$ edges. If $e$ is an edge of $K_{2m}$ regarded as a 2-element subset $\lbrace i,j\rbrace$ of $[2m] = \lbrace 1,2,\dots, 2m\rbrace$, then let $\bar{e}$ denote the complement $[2m]-\lbrace i,j\rbrace$, and let $\bar{M}_i =\lbrace \bar{e}\colon e\in M_i\rbrace$. Color the elements of $\bar{M}_i$ with the color $i$. This gives a proper coloring of the $2m-3$ simplices of $\Delta^{2m-1}$ in $2m-1$ colors.