Diophantine m-tuples for quadratic polynomials

In this paper, we prove that there does not exist a set with more than 98 nonzero polynomials in Z[X], such that the product of any two of them plus a quadratic polynomial n is a square of a polynomial from Z[X] (we exclude the possibility that all elements of such set are constant multiples of a linear polynomial p ∈ Z[X] such that p2|n). Specially, we prove that if such a set contains only polynomials of odd degree, then it has at most 18 elements.