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This question on MathOverflow motivates me to ask what the reasoning is behind the conjecture that there are only finitely many biplanes. More generally, it has been conjectured that for fixed $\lambda>1$ there are only finitely many triples $(v,k,\lambda)$ such that a symmetric balanced incomplete block design with these parameters exists.

My understanding is that prior to the proof that there is no projective plane of order 10, there was some hope that the Bruck-Ryser-Chowla condition would prove to be sufficient for the existence of a design. Yet by that time the finitely-many-biplanes conjecture was already in wide circulation. Since there are infinitely many triples $(v,k,2)$ that satisfy the Bruck-Ryser-Chowla condition, it appears that the design-theory community was able to entertain two conjectures that are not merely incompatible, but starkly opposed.

Is anyone aware of any of the history behind this?

Elaboration (added later): If we think about families of symmetric designs, with $v$ regarded as a parameter and $\lambda$ some function of $v$ (which would make $k$ a function of $v$ as well), then we can talk about "large-$\lambda$" designs, where $\lim_{v\to\infty}\frac{\lambda}{v}=\frac{1}{4}$, and "small-$\lambda$" designs, where $\lim_{v\to\infty}\frac{\lambda}{v}=0$. (Of course there are other possibilities.) Constant $\lambda$, of course, fits into the "small-$\lambda$" case.

There is considerable optimism about "large-$\lambda$" designs. The Hadamard conjecture, for example, would imply that a $(4n-1, 2n-1, n-1)$ exists for every $n$. I think people have conjectured something similar for Menon designs (parameters $(4u^2, 2u^2-u, u^2-u)$), and that most people would make the same conjecture for designs with parameters $(2a^2+2a+1,a^2,a(a-1)/2)$, for which an infinite family is known.

Since my experience is mostly with the "large-$\lambda$" case, I'm curious about the reasons for the extreme pessimism in the "small-$\lambda$" case.

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Will Orrick
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2 Answers2

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I do not think there was ever much hope that Bruck-Ryser-Chowla would be a necessary and sufficient condition for the existence of a symmetric design in general. For projective planes, everything is more strongly constrained, and it was possible to be more optimistic. So it would have been quite reasonable to believe that BRC was sufficient for projective planes, but not for biplanes.

Also it is possible (likely?) that there conjectures were made more in the spirit of "For all I know the following could be true", rather than from a firm conviction based on evidence and deep insight.

Chris Godsil
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  • Thanks for your answer. I'm fairly certain that I've seen older literature where the "BRC is sufficient" conjecture was made in the more general context, but, as you say, this may have been more a speculation about something that might be true than a serious conjecture. I guess it's your statement that there was never much hope that BRC would be sufficient in general that I'm curious about. I have more to say than will fit in this comment, so I've added it to the question. – Will Orrick Jan 12 '13 at 16:39
  • One point is that BRC is a necessary condition for an integer solution, where what you actually want is a 01-solution. Also you should take my answer as a data point; it's what I picked up by osmosis as a grad student. But the people with more knowledge have retired, by and large, so it seemed better to say something. – Chris Godsil Jan 12 '13 at 19:12
  • I've been curious about what the general attitude was before it was proved that there was no plane of order 10, so your remembrances are helpful. By the way, I think you must have meant "rational solution" rather than "integer solution". – Will Orrick Jan 13 '13 at 14:08
  • You're right, I should have rational, not integer. – Chris Godsil Jan 13 '13 at 18:27
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I recently ran across the paper

Daniel Hughes, "Biplanes and semi-biplanes" in Combinatorial Mathematics: Proceedings of the International Conference on Combinatorial Theory, Canberra, August 16 - 27, 1977, Holton, D.A. and Seberry, J. eds., Springer Lecture Notes in Mathematics, Vol. 686, 1978,

which contains the statement

L. J. Dickey and the author have made a computer search for Singer groups for biplanes with $n=k-2\le5000$, using the Honeywell 6060 at the University of Waterloo. We broke the work into several levels, each of which provides additional information about a more restrictive class of biplanes...

The conclusion is:

Theorem 2. If a biplane exists with an abelian Singer group, then its block size $k$ satisfies $k\le9$ or $k\ge5003$.

This might be interpreted as strong evidence for the non-existence of infinitly many biplanes.

Possibly not the earliest such statement to appear in a published work, but it predates the proof of the non-existence of the projective plane of order 10 by more than a decade. (C. W. H. Lam, L. H. Thiel, S. Swiercz, The non-existence of finite projective planes of order 10, Can. J. Math. XVI (1989) 1117–1123.)

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