  We introduce a notion of competitive signaling equilibrium (CSE) in
one-to-one matching markets with a continuum of heterogeneous senders and
receivers. We then study monotone CSE where equilibrium outcomes - sender
actions, receiver reactions, beliefs, and matching - are all monotone in the
stronger set order. We show that if the sender utility is monotone-supermodular
and the receiver's utility is weakly monotone-supermodular, a CSE is stronger
monotone if and only if it passes Criterion D1 (Cho and Kreps (1987), Banks and
Sobel (1987)). Given any interval of feasible reactions that receivers can
take, we fully characterize a unique stronger monotone CSE and establishes its
existence with quasilinear utility functions.
