This paper answers two old questions of Paul Erdős about triple systems. The first asks which finite triple systems are forced to appear inside every triple system whose chromatic number is uncountable. The second asks a related “exact spectrum” question about which uncountable cardinals can occur as the chromatic number of a triple system that avoids a given finite forbidden configuration. The authors give a precise classification and a construction that together settle both problems.

For the classification (Erdős Problem #593) the authors identify a concrete class B of finite triple systems that are exactly the ones forced into every uncountably chromatic triple system. The class B is generated from finite bipartite graphs by a simple operation called a private-vertex expansion (add one new private vertex to each graph edge), together with taking finite disjoint unions and one-point amalgamations (gluing two systems at one vertex). Equivalently, after removing isolated vertices, a finite triple system F is obligatory precisely when F is linear (no two edges share more than one vertex), every hyperedge-node in its Levi graph has an incident bridge (a connection whose removal breaks the Levi graph apart), and every Berge cycle (a cycle that alternates vertices and edges) has even length. These conditions give both a constructive description and an intrinsic graph-theoretic test.

The second main ingredient is an “exact linear calibration” construction. For every uncountable cardinal κ the authors build a linear triple system Lκ whose weak chromatic number equals κ. When κ is the successor cardinal μ+ they can choose Lκ with at most 2^{2^μ} vertices. Combining this construction with the classification gives a clean dichotomy for the exact avoidance spectrum Spec(F) of a finite forbidden triple system F: either no uncountable chromatic numbers occur for F-free systems (exactly when F is in B), or every uncountable cardinal occurs (when F is not in B). This dichotomy settles the three exact-cardinal questions in Erdős Problem #1177 with truth values “yes, no, yes.”