Let $P$ be a property of graphs such that if a graph $G$ has $P$, then any graph obtained from $G$ by removal of a vertex also has $P$.

Let $g(n)$ be the maximum size of a graph of order $n$ having $P$. It can be shown that for $n>2$, $$g(n) \leq \left\lfloor \frac{n}{n-2}\cdot g(n-1)\right\rfloor.$$ I suspect this inequality is well known, and would like to have a reference rather than proving it from scratch. Any pointers will be appreciated!