**DEFINITIONS:** Functions $c : \binom X3\rightarrow \{0\ 1\}$ are called 2-colorings of triangles in $X$. The $4$-element subsets $A\subseteq X$ are called tetrahedra. Each 2-coloring $c$ of triangles induces $(\alpha\ \beta)$-coloring of each tetrahedron $A$, where
$$\beta := \sum_{T\subseteq A,\ |T|=3}\ c(T)\quad\quad\quad \alpha := 4-\beta$$

**QUESTION:** Does every set $X$ admit a 2-coloring $c$ of its triangles such that the induced coloring of every tetrahedron is of the $(2\ 2)$ type? And if the answer is NOT, then what is the smallest cardinality $|X|$ for which $X$ does not admit such $c$ (then such cardinality must be finite)?

**A PARTIAL RESULT:** If $|X| \le 6$ then there exists a 2-coloring of triangles such that all tetrahedra are colored $(2\ 2)$.

(Of course, if such a coloring exists for a set $X$ then it exists--it induces--a similar coloring of triangles in every subset of $X$, hence together with any *good* cardinal number all smaller cardinal numbers are *good* too).