Home » Discrete Geometry » The prehistory of the Sylvester-Gallai theorem

# The prehistory of the Sylvester-Gallai theorem

In 1893 James Joseph Sylvester posed the following problem:

Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line.

This problem was subsequently solved by Tibor Gallai in 1933 and later many solutions appeared. It is nowadays known as the:

Sylvester-Gallai Theorem. Let $\mathcal{S}$ be a finite set of points in the plane. Suppose that the line through any two points of $\mathcal{S}$ passes through a third point of $\mathcal{S}$. Then all the points of $\mathcal{S}$ are collinear.

Much has been written about this theorem and its history. See for example this article. Here I want to speculate about its prehistory. How did Sylvester arrive at this problem? The answer may most likely be found in this figure:

This is a diagram of the affine plane of order 3. There are 9 points and 12 lines. Each line passes through 3 points and each point lies on 4 lines. Four of the lines are not drawn straight. Indeed, it follows from the Sylvester-Gallai Theorem that it is impossible to draw this diagram with all the lines straight. Through any two points there is a line that also passes through a third point.

This combinatorial structure was most likely first discovered by Julius Plücker, and is described in his 1835 book with the title System der analytischen Geometrie, auf neue Betrachtungsweisen gegründet, und insbesondere eine ausführliche Theorie der Curven dritter Ordnung enthaltend. Plücker discovered this structure by studying the inflection points of a nonsingular cubic curve in the complex projective plane.

The cubic curve with homogeneous equation $x^3+y^3+z^3+xyz=0$ has nine inflection points given by homogeneous coordinates

$\mathcal{S} = \{ (0,-1,1), (1, 0, -1), (-1,1, 0),$
$(0,-1,\omega), (\omega, 0, -1), (-1,\omega, 0),$
$(0,-1,\omega^2), (\omega^2, 0, -1), (-1,\omega^2, 0)\}$,

where $\omega$ is a cube root of unity. The set $\mathcal{S}$ has exactly the structure of the affine plane of order 3: the line through any two points of the set contains a third point of the set.

Going back to Sylvester’s wording of his problem, he specifically asks for “real points”. My guess is that he realized that the affine plane of order 3 cannot be realized in the real plane, and that he wondered whether there is any other such structure in the real plane. Interestingly, it does not seem that he had a solution, and the world had to wait another 30 for Paul Erdös to rediscover the question. It is also interesting that Erdös couldn’t solve it either. Gallai’s solution was eventually published in 1944 as one of the solutions to Sylvester’s problem that was posed by Erdös in the American Mathematical Monthly.

Bibliography

1. P. Borwein and W. O. J. Moser, A survey of Sylvester’s problem and its
generalizations
, Aequationes Math. 40 (1990), 111-135.
2. P. Erdös, Problem 4065, Amer. Math. Monthly 50 (1943), 65.
3. P. Erdös, Personal reminiscences and remarks on the mathematical
work of Tibor Gallai
, Combinatorica 2 (1982), 207-212.
4. J. Plücker, System der analytischen Geometrie, auf neue Betrachtungsweisen gegründet, und insbesondere eine ausführliche Theorie der Curven dritter Ordnung enthaltend, Duncker und Humblot, Berlin, 1835.
5. L. M. Pretorius and K. J. Swanepoel, The Sylvester-Gallai theorem, colourings and algebra, submitted. Update: Accepted by Discrete Mathematics.
6. R. Steinberg, Solution to Problem 4065, Amer. Math. Monthly 51 (1944), 169-171.
7. J. J. Sylvester, Mathematical question 11851, Educational Times
59 (1893), 98-99.
8. H. L. de Vries, Historical notes on Steiner systems, Discrete Math. 52 (1984), 293-297.