For r-uniform hypergraphs, rainbow matchings of size at least 2n/(r+1) - O_r(1) are guaranteed, but the maximum guaranteed size is below n - Ω_r(n^{1-1/r}).
Alon, Multicolored matchings in hypergraphs, Moscow Journal of Combinatorics and Number Theory 1 (2011), 3–10
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A note on improved bounds for hypergraph rainbow matching problems
For r-uniform hypergraphs, rainbow matchings of size at least 2n/(r+1) - O_r(1) are guaranteed, but the maximum guaranteed size is below n - Ω_r(n^{1-1/r}).