If a Steiner system S(4,5,17) exists, it would contain derived S(3,4,16) designs. By relying on a recent classification of the S(3,4,16), an exhaustive computer search for S(4,5,17) is carried out. The search shows that no S(4,5,17) exists, thereby ruling out the existence of Steiner systems S(t,t+1,t+13) for t>=4.
['Patric R. J. Östergård', 'Olli Pottonen']
There exists no Steiner system S(4,5,17)