# Sylvester's Problem

The Sylvester Problem has been posed by James Joseph Sylvester in 1893 in Educational Times:

Let $n$ given points have the property that the line joining any two of them passes through a third point of the set. Must the $n$ points all lie on one line?

As H. S. M. Coxeter wrote, "Sylvester himself was doubtless aware of the negative answer in the case of the complex projective plane... But the question as applied to the ordinary real plane remained unanswered for forty years." The answer came as a solution to a problem that P. Erdös - unaware of Sylvester's question - posed in 1943:

1. Let $n$ given points have the property that the straight line joining any two of them passes through a third point of the set. Show that the $n$ points lie on a straight line.
2. Given $n$ points which do not all lie on the same straight line, prove that if we join every two of them we obtain at least $n$ distinct straight lines.

Nowadays, the statement is known as the Sylvester-Gallai theorem, although I have not found a satisfactory explanation for that designation. Four people submitted solutions to Erdös' problem: R. Steinberg, R. C. Buck, T. Grünwald and N. E. Steenrod. The American Mathematical Monthly chose to publish the one by Robert Steinberg and noted that the proposer (Erdös) "enclosed with the problem an outline of Grünwald's solution of part (1) and remarked that part (2) could be easily proved by induction using (1)." (Tibor Grünwald later changed his surname to Gallai.) There is no indication that Steinberg's proof is anyhow related to that of Gallai. Gallai, like Erdös, was a Hungarian mathematician; Steinberg was born in Romania and at the time was a student at the University of Toronto.

One possible reason for the designation that does not credit Steinberg may be the 1948 remark by Coxeter (expanding on the above),

But the question as applied to the ordinary real plane remained unanswered for forty years, and began to seem as intractable as the four-color map problem. Finally, Grünwald established the affirmative answer: such points must all be collinear. Robert Steinberg transformed Grünwald's affine proof into a projective one (this MONTHLY, vol. 51 (1944), p. 169), and L. M. Kelly discovered a still shorter (and quite unrelated) Euclidean proof.

In 1999, this was an established terminology, such that Aigner and Ziegler wrote

Whether Sylvester himself had a proof is in doubt, but a correct proof was given by Tibor Gallai [Grünwald] some 40 years later. Therefore the following theorem is commonly attributed to Sylvester and Gallai. Subsequently to Gallai's proof several others appeared ...

To my count, the difference between the publication dates of the Sylvester (1893) and Erdös (1943) questions is closer to 50 than to 40, so perhaps the reason for the confusion is that Grünwald found and reported his solution to Erdös 10 years before the later decided to go out public with the problem. But the Monthly gave no indication that this was indeed so. In his submission, "Erdös stated that, at Oslo, Karamata asked him about this problem which he had seen stated without proof in an old book about mechanics" with no mention of the date the conversation took place.

The Monthly editors also wrote that "The proof by Buck of part (1) is similar to Grünwald's proof, and he stated that this part of his proof is not original; he was uncertain about the origin of the earlier proof." Still, it remains a mystery whether Gallai ever published his proof prior to sending it as a solution to Erdös' problem.

In 1958, Kelly and Moser declared that "More than sixty years ago, Sylvester proposed the following problem ... An alleged solution (not by Sylvester) advanced at the time proved to be fallacious and the problem remained unsolved until about 1933 when it was revived by Erdös and others." However, all given references were past 1944.

Similarly, in 1970, G. D. Chakerian wrote that "The question was not settled until after 1930, when T Gallai proved an affirmative answer." Again, the only reference that is given is the 1994 solution in the Monthly, where the editors outline the proof in Erdös' submission.

Leaving the question of the proper attribution unsettled, the problem and its solution gave rise to a dispute of an entirely different nature.

According to the editors, it was L. M. Kelly who remarked that part (1) was proposed by Sylvester in the 1893 Educational Times, but Kelly's name is not listed among the solvers. It appears that Kelly has come with a proof of his own some time later. According to Coxeter, Gallai's proof is affine, Steinberg's projective, Kelly's Euclidean. He does not stop at the mere classification but further opines,

It seems to me that parallelism and distance are essentially foreign to this problem, which is concerned only with incidence and order. Thus I would not regard the proofs by Grünwald and Kelly as strictly synthetic. Steinberg's projective proof becomes even more elementary when we transform it into a "descriptive" proof, using the single primitive entity point and the single primitive relation between, in terms of which lines and serial order can be defined.

On another occasion, Coxeter wrote that to apply Euclidean methods to solving the problem of incidence is like using a sledgehammer to crack almonds. But not every one shared this opinion; Kelly's proof - admittedly the shortest among the known ones - has been highlighted as a "proof from the BOOK" by Aigner and Ziegler.

On a historical note, in the last chapter of his Time Travel and Other Mathematical Bewilderments, Martin Gardner describes the "tree planting" or "orchard" problem which is usually "presented with a story about a framer who wishes to plant a certain number $n$ of trees in an orchard so that the pattern of trees will have straight rows of exactly $k$ trees in each row." Gardner wrote that the problem is "related to such mathematical topics as balanced-block design, Kirkman-Steiner triples, finite geometries, Weierstrass elliptic functions, cubic curves, projective planes, error-correcting codes, and many other aspects of significant mathematics." Gardner mentioned that "the mathematician J. J. Sylvester worked continually on the general problem from the late 1960s until his death in 1897." This may suggest the origins of what became known as Sylvester's problem.

### References

1. M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 2000
2. C. Alsina, R. B. Nelsen, Charming Proofs, MAA, 2010
3. P. Borwein, W. O. J. Moser, A survey of Sylvester's problem and its generalizations, Aequationes Mathematicae 40 (1990) 111 - 135
4. G. D. Chakerian, Sylvester's Problem on Collinear Points and a Relative, The American Mathematical Monthly, Vol. 77, No. 2 (Feb., 1970), pp. 164-167
5. H. S. M.Coxeter, A Problem of Collinear Points, The American Mathematical Monthly, Vol. 55, No. 1 (Jan., 1948), pp. 26-28
6. H. S. M.Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
7. P. Erdös, R. Steinberg, Problem 4065 [1943, 65]. Proposed by P. Erdös, Princeton, N. J, Solution by Robert Steinberg, Student, University of Toronto, The American Mathematical Monthly, Vol. 51, No. 3 (Mar., 1944), pp. 169-171
8. M. Gardner, Time Travel and Other Mathematical Bewilderments, W.H.Freeman & Co., 1988.
9. L. M. Kelly, W. O. J. Moser, On the Number of ordinary lines determined by $n$ points, Canad. J. Math. 10(1958), 210-219
10. J. J. Sylvester, Educational Times, Mathematical Question 11851, vol. 59 (1893), p. 98