ex(n, C_{4k+2}^{4-}) equals the number of edges in the complete odd-bipartite 4-graph for large n, and the Turán density of C_{4k+2}^4 is 1/2 for all k≥2 with stability.
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π_link(t) ≤ 1 - t^{-1} - t^{-2}/12 for every t ≥ 2, which determines the order of the gap to the trivial bound 1 - t^{-1} up to a constant factor when paired with Goldwasser's lower bound for prime-power t-1.
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Tur\'an numbers of $4$-uniform tight even cycles minus one edge
ex(n, C_{4k+2}^{4-}) equals the number of edges in the complete odd-bipartite 4-graph for large n, and the Turán density of C_{4k+2}^4 is 1/2 for all k≥2 with stability.
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A note on the $t$-partite link problem of F\"uredi
π_link(t) ≤ 1 - t^{-1} - t^{-2}/12 for every t ≥ 2, which determines the order of the gap to the trivial bound 1 - t^{-1} up to a constant factor when paired with Goldwasser's lower bound for prime-power t-1.