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In this paper, we establish the explicit lower bound estimates for the rankof universal quadratic forms in some certain families of real cubic fieldsunder the condition of density one. The more general results that represent allmultiples of a given rational integer are obtained for totally positivedefinite quadratic lattices. Our main tools are some properties ofindecomposable integers with trace in these fields and short vectors inquadratic lattices.

Lower bound estimates for the rank of universal quadratic forms in some families of real cubic fields with density one