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Sharpening previous results of Baldwin and Shelah [2,11], the authors prove some equivalences to monadic NIP. In particular, they prove that the following are equivalent for a complete theory $T$ with infinite models (see the paper for precise definitions):
\begin{enumerate}[(1)]\item $T$ is monadically NIP, i.e.\ every expansion by unary predicates in NIP;
\item No monadic expansion of $T$ admits coding of pairs of singletons by singletons;
\item $T$ does not admit coding of pairs of tuples by singletons;
\item $T$ has the \emph{f.s.\ dichotomy}: if $\tp(b/Ma)$ is finitely satisfiable in $M \vDash  T$, then for any singleton $c$, either $\tp(b/Mac)$ or $\tp(bc/Ma)$ is finitely satisfiable in $M$;
\item If $M,N \vDash  T$ then any \emph{$M$-f.s.\ decomposition} of any subset of $N$, meaning a decomposition into a sequence of chunks $A_i$ with $\tp(A_i/MA_{<i})$ finitely satisfiable in $M$, extends (in a certain sense) to such a decomposition of $N$;
\item $T$ is dp-minimal and has \emph{indiscernable-triviality}, i.e.\ indiscernability over $B$ is equivalent to indiscernability over each singleton $b \in B$.
\end{enumerate}

They deduce that if $T$ has quantifier elimination in a relational language with finitely many constants, then $T$ is monadically NIP iff $\operatorname{Age}(T)$ is NIP, and they obtain a lower bound on the growth rate of $\operatorname{Age}(T)$ when $T$ is not monadically NIP.
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