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### Abstract

We study oblivious deterministic gossip algorithms for multi-channel radio networks with a malicious adversary. In a multi-channel network, each of the n processes in the system must choose, in each round, one of the c channels of the system on which to participate. Assuming the adversary can disrupt one channel per round, preventing communication on that channel, we establish a tight bound of max $(\Theta(\frac{(1-\epsilon)n}{c-1}+log_cn), \Theta(\frac{n(1-\epsilon)}{\epsilon c^2}))$ on the number of rounds needed to solve the $\epsilon$-gossip problem, a parameterized generalization of the all-to-all gossip problem that requires $(1-\epsilon)n$ of the "rumors" to be successfully disseminated. Underlying our lower bound proof lies an interesting connection between $\epsilon$-gossip and extremal graph theory. Specifically, we make use of Turán's theorem, a seminal result in extremal combinatorics, to reason about an adversary's optimal strategy for disrupting an algorithm of a given duration. We then show how to generalize our upper bound to cope with an adversary that can simultaneously disrupt t < c channels. Our generalization makes use of selectors: a combinatorial tool that guarantees that any subset of processes will be "selected" by some set in the selector. We prove this generalized algorithm optimal if a maximum number of values is to be gossiped. We conclude by extending our algorithm to tolerate traditional Byzantine corruption faults.