On the (6,4)-problem of Brown, Erdős, and Sós

Let f ( r ) ( n ; s , k ) f^{(r)}(n;s,k) be the maximum number of edges of an r r -uniform hypergraph on n n vertices not containing a subgraph with k k edges and at most s s vertices. In 1973, Brown, Erdős, and Sós conjectured that the limit lim n → ∞ n − 2 f ( 3 ) ( n ; k + 2 , k ) \begin{equation*} \lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*} exists for all k k and confirmed it for k = 2 k=2 . Recently, Glock showed this for k = 3 k=3 . We settle the next open case, k = 4 k=4 , by showing that f ( 3 ) ( n ; 6 , 4 ) = ( 7 36 + o ( 1 ) ) n 2 f^{(3)}(n;6,4)=\left (\frac {7}{36}+o(1)\right )n^2 as n → ∞ n\to \infty . More generally, for all k ∈ { 3 , 4 } k\in \{3,4\} , r ≥ 3 r\ge 3 and t ∈ [ 2 , r − 1 ] t\in [2,r-1] , we compute the value of the limit lim n → ∞ n − t f ( r ) ( n ; k ( r − t ) + t , k ) \lim _{n\to \infty } n^{-t}f^{(r)}(n;k(r-t)+t,k) , which settles a problem of Shangguan and Tamo.