  In this paper, we study Einstein gravity extended with Ricci polynomials and
derive the constraints on the coupling constants from the considerations of
being ghost free, exhibiting an $a$-theorem and maintaining causality. The
salient feature is that Einstein metrics with appropriate effective
cosmological constants continue to be solutions with the inclusion of such
Ricci polynomials and the causality constraint is automatically satisfied. The
ghost free and $a$-theorem conditions can only be both met starting at the
quartic order. We also study these constraints on general Riemann cubic
gravities.
