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This work proves a lower bound on longest j-tight cycles in random hypergraphs, suggesting path extensions are crucial for closing cycles.
We prove a lower bound on the length of the longest j-tight cycle in a k-uniform binomial random hypergraph for any 2 ≤ j ≤ k-1. We first prove the existence of a j-tight path of the required length. The standard "sprinkling" argument is not enough to show that this path can be closed to a j-tight cycle - we therefore show that the path has many extensions, which is sufficient to allow the sprinkling to close the cycle.
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Oliver Cooley (2025) studied this question.
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