Olson Constant -- from Wolfram MathWorld

Olson Constant

The Olson constant of a finite Abelian group is the least integer such that every -element subset of has a nonempty subset summing to zero. Thus it is one more than the largest size of a zero-sum-free subset. Its elements must be distinct, unlike the repeated terms allowed in the definition of the Davenport constant (Olson 1975, Hamidoune and Zémor 1996).

For cyclic groups with , 2, ..., the first values are 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, .... They are one more than the entries of OEIS A034463, whose indexing starts at and whose values count the maximum zero-sum-free cardinality. For a prime , Balandraud (2012) proved, with the minimum over positive integers ,

$Ol(C_p)=min{k>=1:k(k+1)/2>=p}.$

The same formula does not hold for all composite orders.

For a general finite Abelian group, Liu (2026) proved

More precisely, a zero-sum-free -element subset has at least distinct nonempty subset sums. This improves the error term in the classical square-root estimate related to the already solved Erdős problem 540.

References

Balandraud, E. "An Addition Theorem and Maximal Zero-Sum Free Sets in

." Israel J. Math. 188, 405-429, 2012. https://doi.org/10.1007/s11856-011-0171-9.

Hamidoune, Y. O. and Zémor, G. "On Zero-Free Subset Sums." Acta Arith. 78, 143-152, 1996. https://doi.org/10.4064/aa-78-2-143-152.

Liu, C. "A Logarithmic Error Term for Subset Sums of Zero-Sum-Free Sets." 28 Oct 2026. https://arxiv.org/abs/2610.38373.

Olson, J. E. "Sums of Sets of Group Elements." Acta Arith. 28, 147-156, 1975. https://doi.org/10.4064/aa-28-2-147-156.

Sloane, N. J. A. Sequence A034463 in "The On-Line Encyclopedia of Integer Sequences."

Olson Constant

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