Subalgebras of Golod-Shafarevich Algebras

A graded associative algebra generated by m elements of degree one is called Golod–Shafarevich (GS) if it is presented with less than m2/4 relators of degree at least two. We explore conditions under which subalgebras of graded GS algebras are themselves GS. We prove that infinitely many Veronese powers of an algebra presented by m generators and r relators are GS if $r \frac{4}{25}m^2$.