Shannon capacity of large odd cycles

It is known that lim inf/sub n/spl rarr//spl infin//((2n+1)/2-/spl Theta/(C/sub 2n+1/))=0 and that the lim sup of this difference is at most 1/4, where /spl Theta/(G) is the Shannon (1956) capacity of the graph G. We prove that the above lim sup is at most 1/6 and conjecture that the limit itself exists and equals 0. We show that the lim sup is small by constructing large independent sets in the third power of C/sub 2n+1/.