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 ) {equation*} lim n→ ∞ n⁻² f⁽³⁾(n;k+2,k) {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)= (7/36+o(1) )n^2 as n → ∞ n→ ∞ . More generally, for all k ∈ { 3 , 4 } k∈ \{3,4\} , r ≥ 3 r≥ 3 and t ∈ [ 2 , r − 1 ] t∈ [2,r-1] , we compute the value of the limit lim n → ∞ n − t f ( r ) ( n ; k ( r − t ) + t , k ) lim n→ ∞ n⁻ᵗf⁽ʳ⁾(n;k(r-t)+t,k) , which settles a problem of Shangguan and Tamo.