Ackerman 2008 there are not too many magic configurations

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theorem_1: Ackerman, Buchin, Knauer, Pinchasi and Rote's proof of Murty's conjecture that a finite planar point set with positive weights summing to 1 on every determined line has all but at most one point collinear, has no three points collinear, or is projectively the 7-point failed Fano configuration.

theorem_2: The paper's reduction target: two nonempty planar point sets whose union has as ordinary lines exactly the lines through two points of the first set, and carries positive weights summing to 1 on every determined line, form projectively the failed Fano configuration, the first set being the points of weight one half.

theorem_4: Shows that two nonempty disjoint planar point sets, the second in general position, with no line determined by the first and no ordinary line of the union through a point of the second, form projectively the failed Fano configuration.

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Ackerman, Eyal and Buchin, Kevin and Knauer, Christian and Pinchasi, Rom and Rote, Günter, There are not too many magic configurations. Discrete Comput. Geom. 39 (2008), 3-16. DOI 10.1007/s00454-007-9023-0.

A finite planar point set P is a magic configuration if positive weights can be assigned to its points so that on every line determined by P the weights sum to 1. Theorem 1 (p. 1) proves Murty's 1971 conjecture: a magic configuration of n points has n - 1 (or n) collinear points, or is in general position with no three points collinear, or is a 7-point configuration that up to projective transformation is the failed Fano configuration of Figure 1 (p. 2), whose weights the figure shows. The paper proves only this direction. The reduction on pp. 1-2 uses the Gallai-Sylvester theorem and the Kelly-Moser bound of at least 3(n-1)/7 ordinary lines for the n - 1 points left after deleting one point to show that every point on an ordinary line has weight 1/2, and passes to Theorem 2 (p. 2): if A and B are nonempty point sets, the ordinary lines of A union B are exactly the lines through two points of A, and positive weights sum to 1 on every determined line, then A union B is projectively the failed Fano configuration with A the points of weight 1/2. Theorem 2 is proved through its dual on great circles of a sphere, Theorem 3 (p. 3), by a discharging argument (Section 2, pp. 3-10). Section 3 (pp. 10-11) records Theorem 4 (p. 10), a version of Theorem 2 without weights when B is in general position, applies it to geometrically induced perfect matchings, and notes that any weights witnessing a magic configuration are unique.

Pages and labels on this card and its result pages are those of the manuscript named below.

Source: https://page.mi.fu-berlin.de/rote/Papers/allpapers.html#There+are+not+too+many+magic+configurations. The copy read for this card is the author's manuscript dated February 27, 2007, which prints no notice, from the author's publications page (https://page.mi.fu-berlin.de/rote/Papers/allpapers.html), which states no terms; the term is unstated.

Bears on. #735: the problem asks when n points can be given positive weights with the same sum on every line through at least two of them; the paper's magic configurations fix that sum at 1, and Theorem 1 (p. 1) lists the only configurations that can be magic. The paper shows the weights for the failed Fano configuration and does not state the converse for the other two families.

Results.

• Theorem 1 (p. 1): every magic configuration of n points has n - 1 (or n) collinear points, is in general position, or is projectively the 7-point failed Fano configuration.
• Theorem 2 (p. 2), with its dual Theorem 3 (p. 3): for nonempty point sets A and B, if the ordinary lines of A union B are exactly the lines through two points of A and positive weights sum to 1 on each determined line, then A union B is projectively the failed Fano configuration, with A the points of weight 1/2.
• Theorem 4 (p. 10): for nonempty disjoint point sets A and B with B in general position, if no line determined by A and no ordinary line of A union B passes through a point of B, then A union B is projectively the failed Fano configuration.

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