Fisher's Block Design Inequality

A balanced incomplete Block Design ($v$, $k$, $\lambda$, $r$, $b$) exists only for $b\geq v$ (or, equivalently, $r\geq k$).

See also Bruck-Ryser-Chowla Theorem

References

Dinitz, J. H. and Stinson, D. R. ``A Brief Introduction to Design Theory.'' Ch. 1 in Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, pp. 1-12, 1992.