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The ability of a P2P network to scale its throughput up in proportion to the arrival rate of peers has recently been shown to be crucially dependent on the chunk sharing policy employed. Some policies can result in low frequencies of a particular chunk, known as the missing chunk syndrome, which can dramatically reduce throughput and lead to instability of the system. For instance, commonly used policies that nominally ``boost'' the sharing of infrequent chunks such as the well-known rarest-first algorithm have been shown to be unstable. We take a complementary viewpoint, and instead consider a policy that simply prevents the sharing of the most frequent chunk(s), that we call mode-suppression. We also consider a more general version that suppresses the mode only if the mode frequency is larger than the lowest frequency by a fixed threshold. We prove the stability of mode-suppression using Lyapunov techniques, and use a Kingman bound argument to show that the total download time does not increase with peer arrival rate. We then design versions of mode-suppression that sample a small number of peers at each time, and construct noisy mode estimates by aggregating these samples over time. We show numerically that mode suppression stabilizes and outperforms all other recently proposed chunk sharing algorithms, and via integration into BitTorrent implementation operating over the ns-3 that it ensures stable, low sojourn time operation in a real-world setting.