New turan exponents for two extermal hypergraph problems

An r-uniform hypergraph is called t-cancellative if for any t + 2 distinct edges A1, . . . ,At,B,C, it holds that (∪ ti =1Ai) ∪ B not = (∪ t i=1Ai) ∪ C. It is called t-union-free if for any two distinct subsets scrA; , scrB , each consisting of at most t edges, it holds that ∪ Ain scrA; A not = ∪ Bin scrB B. Let Ct(n, r) (resp., Ut(n, r)) denote the maximum number of edges of a t-cancellative (resp., t-union-free) r-uniform hypergraph on n vertices. Among other results, we show that for fixed r geq 3, t geq 3 and n rightarrow infty , ω (nlfloor 2r t+2 rfloor; +2r ({m}{o}{d} t+2) t+1 ) = Ct(n, r) = O(n lceil r lfloor t/2⌋ +1 ⌉ ) and Ω (n r t 1 ) = Ut(n, r) = O(nlceil r t 1 ⌉ ), thereby significantly narrowing the gap between the previously known lower and upper bounds. In particular, we determine the Turán exponent of Ct(n, r) when 2 | t and (t/2 + 1) | r, and of Ut(n, r) when (t 1) | r. The main tool used in proving the two lower bounds is a novel connection between these problems and sparse hypergraphs.