Integers $n$ with $m+\omega(m)\le n$ for all $m<n$: are there infinitely many? (Erdős #413)

Let $\omega(m)$ be the number of **distinct** prime factors of $m$ (with $\omega(1)=0$). Call $n$ a *champion* if $m+\omega(m)\le n$ for every $1\le m<n$; equivalently $\omega(n-j)\le j$ for all $1\le j<n$. Are there infinitely many champions? More strongly, is there a fixed $\epsilon>0$ with infinitely many $n$ such that $m+\epsilon\,\omega(m)\le n$ for all $m<n$?

Acceptance. PARTIAL / EXTENDS: enumerate champions $n$ up to a large bound (checking $\omega(n-j)\le j$ via a sieve for $\omega$), tabulate them and estimate their density/growth; separately, for a fixed small $\epsilon$, find $n$ with $m+\epsilon\,\omega(m)\le n$ for all $m<n$. FULLY RESOLVES: a proof the champion set (or its $\epsilon$-strengthening) is infinite. Provide the sieve and the enumerated list.