Lower bounds on mt(r, s)

Useful lower bounds are obtained for m_t(r, s), s a prime power, r ? t ? 3, by relating (t ? 1)-flats of PG(r ? 1, s) to (t ? 1) and lower dimensional flats of its subgeometries. Explicit expressions in general are obtained for these bounds when t = 3. A general method is developed for deriving these bounds when t > 3.