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Let χl(G) denote the list chromatic number of the r‐uniform hypergraph G. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if G is simple and d‐regular, then χl(G)≥(1/(r−1)+o(1))logrd. To see how close this inequality is to best possible, we examine χl(G) when G is a random r‐partite hypergraph with n vertices in each class. The value when r = 2 was determined by Alon and Krivelevich; here we show that χl(G)=(g(r,α)+o(1))logrd almost surely, where d is the expected average degree of G and α=lognd. The function g(r,α) is defined in terms of "preference orders" and can be determined fairly explicitly. This is enough to show that the container method gives an optimal lower bound on χl(G) for r = 2 and r = 3, but, perhaps surprisingly, apparently not for r ≥ 4. [ABSTRACT FROM AUTHOR]

Let R(Clr,Knr) be the Ramsey number of an r‐uniform loose cycle of length l versus an r‐uniform clique of order n. Kostochka et al. showed that for each fixed r≥3, the order of magnitude of R(C3r,Knr) is n3∕2 up to a polylogarithmic factor in n. They conjectured that for each r≥3 we have R(C5r,Knr)=O(n5∕4). We prove that R(C53,Kn3)=O(n4∕3), and more generally for every l≥3 that R(Cl3,Kn3)=O(n1+1∕⌊(l+1)∕2⌋). We also prove that for every l≥5 and r≥4, R(Clr,Knr)=O(n1+1∕⌊l∕2⌋) if l is odd, which improves upon the result of Collier‐Cartaino et al. who proved that for every r≥3 and l≥4 we have R(Clr,Knr)=O(n1+1∕(⌊l∕2⌋−1)). [ABSTRACT FROM AUTHOR]