Vertex counts and intersections in 3-chromatic hypergraphs

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evidence/: Exact mathematical extracts, independent reports and separate grades for the vertex-count and linear-intersection arguments. Both warrants are noncomputational.

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For every integer $r\ge2$, an intersecting $r$-uniform hypergraph with no proper $2$-coloring has two distinct edges meeting in at least $\lfloor r/2\rfloor$ vertices. The result includes infinite hypergraphs in ordinary classical set theory with choice, and implies the uniform bound $r/3$. It answers the second question of Problem 836 with chromatic number exactly $3$.

The independently examined argument and separate grade retain the complete compactness, recoloring and counting derivations. They also retain a two-edge family showing that mere proper $3$-colorability does not imply a uniform linear intersection bound. The mathematical result is credited on Price's claim page; the reconstruction and independent examination are Plasma AI's work.

The first question has a separate independent examination and passing grade. For every integer $r\ge2$, the Erdős–Lovász construction is finite, intersecting, $r$-uniform and exactly $3$-chromatic, has no isolated vertices, and has $2r-2+\tfrac12\binom{2r-2}{r-1}$ vertices. Dividing by $r^2$ tends to infinity, so no uniform quadratic bound holds. Its pending source claim retains the historical attribution and review scope.

The first construction supplies finite counterexamples, sufficient for the negative answer even when arbitrary hypergraphs are allowed. The second argument applies to arbitrary non-2-colorable hypergraphs using classical choice. Thus the two reviewed statements cover both modified questions with the same nonmonochromatic-edge coloring convention. The stronger second statement needs no exact-3 upper bound. Each review retains its own independence limits and excludes fidelity to an external source document and all external Lean artifacts.