On extensions of $D(4)$-triples by adjoining smaller elements

This paper examines the problem of obtaining a $D(4)$-quadruple by adding a smaller element to a $D(4)$-triple. We prove some relations between elements of observed hypothetical $D(4)$-quadruples under which a conjecture of the uniqueness of such an extension holds. Also, it is shown that for any $D(4)$-triple, there are at most two extensions with a smaller element.