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Towards Ryser’s conjecture. (English) Zbl 1030.05018

Casacuberta, Carles (ed.) et al., 3rd European congress of mathematics (ECM), Barcelona, Spain, July 10-14, 2000. Volume I. Basel: Birkhäuser. Prog. Math. 201, 533-541 (2001).
Summary: Ryser’s conjecture asserts that there is no \((v, k,\lambda)\)-difference set with \(\text{gcd}(v, k -\lambda)>1\) in any cyclic group. We survey what is known on this conjecture and obtain progress towards it by improving the exponent bound for difference sets in [B. Schmidt, J. Am. Math. Soc. 12, 929-952 (1999; Zbl 0939.05016)]. As a consequence, with three possible exceptions, Ryser’s conjecture is true for all parameters of known \((v, k,\lambda)\)-difference sets with \(k\leq 5\cdot 10^{10}\). In particular, the circulant Hadamard matrix conjecture holds for orders \(\leq 10^{11}\), also with only three possible exceptions. Finally, we obtain the first necessary and sufficient condition known in the literature for the existence of an infinite class of difference sets not relying on the self-conjugacy assumption.
For the entire collection see [Zbl 0972.00031].

MSC:

05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)

Citations:

Zbl 0939.05016
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