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We investigate the upper chromatic number of the hypergraph formed by thepoints and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, themost number of colors that can be used to color the points so that every$k$-subspace contains at least two points of the same color. Clearly, if onecolors the points of a double blocking set with the same color, the rest of thepoints may get mutually distinct colors. This gives a trivial lower bound, andwe prove that it is sharp in many cases. Due to this relation with doubleblocking sets, we also prove that for $t\leq \frac38p+1$, a small $t$-fold(weighted) $(n-k)$-blocking set of $\mathrm{PG}(n,p)$, $p$ prime, must containthe weighted sum of $t$ not necessarily distinct $(n-k)$-spaces.

On the upper chromatic number and multiplte blocking sets of PG($n,q$)