Factors and loose Hamilton cycles in sparse pseudorandom hypergraphs We investigate the emergence of spanning structures in sparse pseudorandom kuniform hypergraphs, using the following comparatively weak notion of pseudorandomness. A kuniform hypergraph H on n vertices is called (p,alpha,epsilon)pseudorandom if for all (not necessarily disjoint) vertex subsets A_1,dots, A_ksubseteq V(H) with A_1cdots A_kgeqalpha nk we have e(A_1,dots, A_k)(1pmepsilon)p A_1cdots A_k. For any linear kuniform F we provide a bound on alphaalpha(n) in terms of pp(n) and F, such that (under natural divisibility assumptions on n) any kuniform big(p,alpha, o(1)big)pseudorandom nvertex hypergraph H with a mild minimum vertex degree condition contains an Ffactor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudorandom hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudorandomness such as jumbledness or large spectral gap.  a :129,consequence, big(p,alpha, o(1)big)pseudorandom kgraphs as above contain: (i) a perfect matching if alphao(pk) and (ii) a loose Hamilton cycle if alphao(pk1). This extends the works of LenzMubayi, and LenzMubayiMycroft who studied the analogous problems in the dense setting.

Extremal Problems for tPartite and tColorable Hypergraphs Fix integers t ge r ge 2 and an runiform hypergraph F. We prove that the maximum number of edges in a tpartite runiform hypergraph on n vertices that contains no copy of F is c_t, Fn choose r o(nr), where c_t, F can be determined by a finite computation.   We explicitly define a sequence F_1, F_2, ldots of runiform hypergraphs, and prove that the maximum number of edges in a tchromatic runiform hypergraph on n vertices containing no copy of F_i is alpha_t,r,in choose r o(nr), where alpha_t,r,i can be determined by a finite computation for each ige 1. In several cases, alpha_t,r,i is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.