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Dec 31, 2013 at 0:01 comment added Dustin G. Mixon Wow, you're right. I ignored a log factor in $n$ on the right-hand side, thinking I didn't need it, but I do. For the sake of documentation, I found a lot of useful information by googling "graph discrepancy".
Dec 30, 2013 at 23:15 comment added Terry Tao Such graphs cannot be arbitrarily large (for fixed $C$), because any sufficiently large graph of a given density will contain a copy of the complete bipartite graph $K_{m,m}$ for any fixed $m$, which will contradict your property for $m$ large enough depending on $C$. (Graphs obeying your property will have density about 1/2.)
Dec 30, 2013 at 15:24 history asked Dustin G. Mixon CC BY-SA 3.0