The exclusive production cross sections for $omega$ and $phi$ mesons have been measured in proton-proton reactions at $p_{lab}=3.67$ GeV/c. The observed $phi/omega$ cross section ratio is $(3.8pm0.2^{+1.2}_{-0.9})times 10^{-3}$. After phase space corrections, this ratio is enhanced by about an order of magnitude relative to naive predictions based upon the Okubo-Zweig-Iizuka (OZI) rule, in comparison to an enhancement by a factor $sim 3$ previously observed at higher beam momenta. The modest increase of this enhancement near the production threshold is compared to the much larger increase of the $phi/omega$ ratio observed in specific channels of $bar pp$ annihilation experiments. Furthermore, differential cross section results are also presented which indicate that although the $phi$ meson is predominantly produced from a $^3P_1$ proton-proton entrance channel, other partial waves contribute significantly to the production mechanism at this beam momentum.
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