For a given subset S of positive integers, a positive-definite integral quadratic form is called S-universal if it represents every integer in the set S. We say that an S-universal form has minimal dimension if there are no S-universal forms of a lower dimension. The goal of this paper is to study the size of the bound of the discriminant of positive-definite integral S-universal quadratic forms of minimal dimension in the case when S is a finite subset of positive integers.
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