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If the dark matter (DM) is non-relativistic at freeze-out and if the kinetic decoupling occurs much later than freeze-out, it is called cold dark matter. More precisely, it is stated that if the DM were not non-relativistic at $\underline{1~ \text{KeV}}$ it would wash out structures $\underline{\text{down to every scale}}$.

$\bullet$ Where does this number $1$ KeV come from?

$\bullet$ Why we say we observe structures down to every possible scale (for example, see page 8 of this note) what is the smallest scale are we thinking of?

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Basically, if the dark matter is semi-relativistic, the temperature of the dark matter means it has enough kinetic energy in the early universe to escape small gravitational wells, which means that any initial overdensities on small scales are washed out. This is called "free streaming".

The number 1 KeV (or some argue more like 4 KeV) are the observational bounds on thermally produced (warm) dark matter mass. Basically, if the dark matter was lighter than this, free streaming would suppress structure formation on larger scales (which we have observed not to be such suppressed).

We do not observe "every possible scale", what the note writes is that we don't see any of the suppression from free-streaming on the scales that we actually can observe. Our current limits are dark matter halo masses of something like $10^9 -10^{10} M_\odot$.

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