== 0. self-test of the THC counters ==
complete k-graphs give (n-1)! for both counters; 300 random k-graphs: counters agree (254 of them have a THC)

== 1. hemi-icosahedron facts ==
(F0) D': pair multiplicities [2] (every pair of W in exactly 2 blocks)
(F0) D: pair multiplicities [2] (every pair of W in exactly 2 blocks)
(F1) T in D' => W\T in D : True   (hence neither D nor D' contains a complementary pair)
D' as a 2-complex: V=6, E=15, F=10, chi=1; all vertex links are 5-cycles: True  => 6-vertex RP^2 triangulation
labelled 2-(6,3,2) designs on 6 points: 12; isomorphism classes: 1; |Aut(D')| = 60; 720/|Aut| = 12
(F2) D': all 60 ordered blocks: forced walk x1..x5 distinct and x6 = x1: True; walk is 5-periodic: True; sequences of 6 distinct points with 4 consecutive blocks: 0
(F2) D: all 60 ordered blocks: forced walk x1..x5 distinct and x6 = x1: True; walk is 5-periodic: True; sequences of 6 distinct points with 4 consecutive blocks: 0

== 2. H_3 and the family H_k ==
H_3 (apex + 15 apex triples + 10 faces): |E| = 25, codegree histogram {3: 15, 5: 6}, min codegree 3 vs floor((7-3+3)/2) = 3
   THC sequences: counter A 0, counter B 0, all 7! = 5040 orders unrestricted: 0  [0.01s]
   orders w1..w6 of W with w1w2w3, w2w3w4, w3w4w5, w4w5w6 all faces: 0
   |Aut(H_3)| = 60, all fix the apex: True, element orders {1: 1, 2: 15, 3: 20, 5: 24} (A_5: {1:1, 2:15, 3:20, 5:24})
H_3: n=7, |E|=25, codegree hist {3: 15, 5: 6}, min 3 vs floor((n-k+3)/2) = 3; THC sequences: A 0, B 0  [0.0s]
H_4: n=8, |E|=50, codegree hist {3: 40, 5: 16}, min 3 vs floor((n-k+3)/2) = 3; THC sequences: A 0, B 0  [0.0s]
H_5: n=9, |E|=96, codegree hist {3: 75, 5: 51}, min 3 vs floor((n-k+3)/2) = 3; THC sequences: A 0, B 0  [0.1s]
H_6: n=10, |E|=170, codegree hist {3: 120, 5: 132}, min 3 vs floor((n-k+3)/2) = 3; THC sequences: A 12960, B 12960  [1.0s]  witness (0, 1, 2, 3, 6, 7, 4, 8, 9, 5)
H_7: n=11, |E|=280, codegree hist {3: 175, 5: 287}, min 3 vs floor((n-k+3)/2) = 3; THC sequences: A None, B 259200  [4.9s]  witness (0, 1, 2, 3, 6, 7, 4, 8, 9, 10, 5)
   (for k = 6, 7 the construction has min codegree 3 but a THC exists; the family stops at k = 5, as claimed)

   hand-proof lemma: every placement of the k-2 apexes forces (i) a complementary pair or (ii) four
   consecutive W-triples to lie in D (4-windows with exactly one apex must have their W-part in D)
   k=3: all 6 apex placements (w_1 at position 0) satisfy (i) or (ii)
   k=4: all 21 apex placements (w_1 at position 0) satisfy (i) or (ii)
   k=5: all 56 apex placements (w_1 at position 0) satisfy (i) or (ii)

== 4. the (3,9) and (5,10) examples (claimant's files, re-checked) ==
example_k3_n9.txt: n=9, k=3, |E|=50, vertex degrees 16..18, codegree hist {4: 33, 6: 3}; min 4; floor((n-k+3)/2) = 4, (n-k+3)/2 = 4.5
   THC sequences: counter B 0 [0.0s], counter A over all 40320 sequences 0 [0.0s]
example_k5_n10.txt: n=10, k=5, |E|=191, vertex degrees 95..96, codegree hist {4: 130, 5: 45, 6: 35}; min 4; floor((n-k+3)/2) = 4, (n-k+3)/2 = 4.0
   THC sequences: counter B 0 [0.2s], counter A over all 362880 sequences 0 [0.5s]
   min codegree 4 >= (n-k+3)/2 = 4 exactly: refutes the version without the floor too

== 5. exhaustive check for n <= 6 (all k with 3 <= k <= n-1) ==
k=3, n=4: threshold 2; k-graphs meeting it: 1; without THC: 0  (distinct THC edge-sets: 1)
k=3, n=5: threshold 2; k-graphs meeting it: 26; without THC: 0  (distinct THC edge-sets: 12)
k=3, n=6: threshold 3; k-graphs meeting it: 271; without THC: 0  (distinct THC edge-sets: 60)
k=4, n=5: threshold 2; k-graphs meeting it: 1; without THC: 0  (distinct THC edge-sets: 1)
k=4, n=6: threshold 2; k-graphs meeting it: 76; without THC: 0  (distinct THC edge-sets: 60)
k=5, n=6: threshold 2; k-graphs meeting it: 1; without THC: 0  (distinct THC edge-sets: 1)

[total 8.2s]
