the default approach seems to be the Bron–Kerbosch algorithm

From the complexity estimates given there, a worst case of at least 3**(1000/3) = 7.609e+158 for cliques of size 1000 is to be expected. °

I.o.W. you should already reserve a table at the The Restaurant at the End of the Universe to take a break in between.

Another approach would be porting Perl to quantum computers and hoping that Google and IBM are producing more than vapor ware in your lifetime.

update Nov 29

°) for the vertex-ordering version

from Degeneracy_(graph_theory)#Relation_to_other_graph_parameters

A k-degenerate graph has chromatic number at most k + 1; this is proved by a simple induction on the number of vertices which is exactly like the proof of the six-color theorem for planar graphs. Since chromatic number is an upper bound on the order of the maximum clique, the latter invariant is also at most degeneracy plus one.