The limit in the (𝑘+2,𝑘)-problem of Brown, Erdős and Sós exists for all 𝑘≥2

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 positive integers k ≥ 2 k\ge 2 . They proved this for k = 2 k=2 . In 2019, Glock proved this for k = 3 k=3 and determined the limit. Quite recently, Glock, Joos, Kim, Kühn, Lichev and Pikhurko proved this for k = 4 k=4 and determined the limit; we combine their work with a new reduction to fully resolve the conjecture by proving that indeed the limit exists for all positive integers k ≥ 2 k\ge 2

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