Geodesic
The No-Three-In-Line Problem
Canadian mathematical bulletin, Tome 11 (1968) no. 4, pp. 527-531

Voir la notice de l'article provenant de la source Cambridge University Press

Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that 
Guy, Richard K.; Kelly, Patrick A. The No-Three-In-Line Problem. Canadian mathematical bulletin, Tome 11 (1968) no. 4, pp. 527-531. doi: 10.4153/CMB-1968-062-3
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