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$2$-colorings of triangles, resulting in $(2\ 2)$-colorings of all tetrahedra

When $|X|=7$, no such coloring exists. In this case there are 35 triangles and 35 tetrahedra. If $A$ is a tetrahedron, let $(\alpha_A,\beta_A)$ be its type. Each triangle in $X$ is a subset of exact …
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