In this paper we give sublinear-time distributed algorithms in the CONGEST\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {CONGEST}$$\end{document} model for finding or listing cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be detected and listed in sublinear time, O(n5/6+o(1))\documentclass[12pt]{minimal}
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\begin{document}$$O(n^{5/6+o(1)})$$\end{document} rounds and O(n73/75+o(1))\documentclass[12pt]{minimal}
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\begin{document}$$O(n^{73/75+o(1)})$$\end{document} rounds, respectively. For even-length cycles, C2k\documentclass[12pt]{minimal}
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\begin{document}$$C_{2k}$$\end{document}, we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from O~(n5/6)\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{O}}(n^{5/6})$$\end{document} to O~(n3/4)\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{O}}(n^{3/4})$$\end{document} rounds. We also show two obstacles on proving lower bounds for C2k\documentclass[12pt]{minimal}
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\begin{document}$$C_{2k}$$\end{document}-freeness: first, we use the new connection to extremal combinatorics to show that the current lower bound of Ω~(n)\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{\varOmega }}(\sqrt{n})$$\end{document} rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant δ∈(0,1/2)\documentclass[12pt]{minimal}
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\begin{document}$$\delta \in (0,1/2)$$\end{document} such that for anyk, a lower bound of Ω(n1/2+δ)\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega (n^{1/2+\delta })$$\end{document} on C2k\documentclass[12pt]{minimal}
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\begin{document}$$C_{2k}$$\end{document}-freeness would imply new lower bounds in circuit complexity. We use the same technique to show a barrier for proving any polynomial lower bound on triangle-freeness. For general subgraphs, it was shown by Fischer et al. that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Ω~(n2-Θ(1/k))\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{\varOmega }}(n^{2-\varTheta (1/k)})$$\end{document} rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in O(n2-Θ(1/k))\documentclass[12pt]{minimal}
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\begin{document}$$O(n^{2 - \varTheta (1/k)})$$\end{document} rounds, nearly matching the lower bound.
