 Solution

The compactness theorem states that a set of first-order sentences has a model if every finite subset has a model.

For each $n\ge 1$, let ${\chi }_n$ assert that at least $n$ distinct elements exist. Consider $T=\{\varphi \}\cup \{{\chi }_n:n\ge 1\}$. A finite subset requires only some finite lower bound $N$ on the size. Choose an odd integer $m\ge N$; the given property of $\varphi$ supplies a model of that size, satisfying the finite subset. By the compactness theorem, $T$ has a model. Its domain has at least $n$ elements for every $n$, so $\varphi$ has an infinite model. This argument applies to every sentence with the finite-model-size property in (b), irrespective of its behavior on infinite structures.