DVR-valued points of schemes

Let $X$ be a scheme of finite type over a discrete valuation ring $R$ with fraction field $K$, such that the generic fibre $X_K$ is smooth over $K$. Let $Y$ be a closed subscheme of $X$ which contains no irreducible component of $X$.

Is it true - maybe under some additional assumptions on $X$ and/or $R$ - that $X(R) \setminus Y(R)$ is dense in $X(R)$ for the natural topology defined by $R$? If so, is there an easy way to see this or does it require heavy machinery?

Also, do there exist simple counterexamples when $X_K$ is not smooth over $K$?

2022-07-25 20:42:46
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