Saturday, 8 September 2007

graph theory - How to estimate the growth of the probability that G(n,M)G(n,M) contains a kk-clique

You might take a look at Chapter VII of Bollobas. In particular,
Theorem VII.1.7 -- which is simple enough that he doesn't bother providing a proof -- states that the expected number of kk-cliques in G(n,M)G(n,M) is, setting N=nchoose2N=nchoose2 and K=kchoose2K=kchoose2,
nchooseknKchooseMKNchooseM1.nchooseknKchooseMKNchooseM1.
Also, Theorem VII.3.7 states that if M=o(n2/(k1))M=o(n2/(k1)) then with probability tending to one, Gn,MGn,M contains no kk-clique, whereas if M/n2/(k1)toinftyM/n2/(k1)toinfty then with probability tending to one Gn,MGn,M does contain a kk-clique. I know this doesn't fully answer your question but it may help.



Incidentally, (you probably already realize that) it is a priori possible (though I don't think it is the case) that, for example, tk(M+1)tk(M)geqfrac1mathrmpoly(n)tk(M+1)tk(M)geqfrac1mathrmpoly(n) for all kchoose2leqMleqlceilfrac(k1)Nkrceilkchoose2leqMleqlceilfrac(k1)Nkrceil, since all we really know by Turán is that
sumlceil(k1)N/krceilM=K(tk(M+1)tk(M))=1.sumlceil(k1)N/krceilM=K(tk(M+1)tk(M))=1.

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