A note on one-variable theorems for NSOP

We give an example of an SOP theory $T$, such that any $L(M)$-formula $\phi (x,y)$ with $\mathrm{\mid }y\mathrm{\mid }=1$ is NSOP. We show that any such $T$ must have the independence property. We also give a simplified proof of Lachlan’s theorem that if every $L$-formula $\phi (x,y)$ with $\mathrm{\mid }x\mathrm{\mid }=1$ is NSOP, then $T$ is NSOP.