On the $(k+2,k)$-problem of Brown, Erdős and Sós for $k=5,6,7$

TL;DR

This work resolves the Brown–Erdős–Sós quadratic-growth regime for the degenerate hypergraph Turán problem in the cases k ∈ {5,6,7} across all uniformities r ≥ 3. It combines high-girth construction techniques with a robust merging/weighting framework to determine exact limiting values for f^{(r)}(n; rk−2k+2,k): the limits for k ∈ {5,7} equal 1/(r^2−r−1) for all r≥3, while k=6 exhibits a split behavior with 61/330 for r=3 and 1/(r^2−r) (r≥4). A unified lower-bound approach via packing dense G_k^{(r)}-free bases and an intricate upper-bound strategy based on cluster weightings and merging show the limits match, yielding precise asymptotics. The results also feed into asymptotics for generalized Ramsey numbers, linking degenerate hypergraph Turán theory to Ramsey-type thresholds. These advances illuminate the structure of extremal 3- and higher-uniform hypergraphs at quadratic growth and point to further parameter regimes for future work.

Abstract

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no subgraph with $k$ edges and at most $s$ vertices. Brown, Erdős and Sós [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press 1973] conjectured that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. The value of the limit was previously determined for $k=2$ in the original paper of Brown, Erdős and Sós, for $k=3$ by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for $k=4$ by Glock, Joos, Kim, Kühn, Lichev and Pikhurko [Proc. Amer. Math. Soc., Series B, 11 (2024) 173-186] while Delcourt and Postle [Proc. Amer. Math. Soc., 152 (2024), 1881-1891] proved the conjecture (without determining the limiting value). In this paper, we determine the value of the limit in the Brown-Erdős-Sós Problem for $k\in \{5,6,7\}$. More generally, we obtain the value of $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all $r\geq 3$ and $k\in \{5,6,7\}$. In addition, by combining these new values with recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new asymptotic values for several generalised Ramsey numbers.

Figures (3)

• Figure 1: Examples of $\mathcal{S}_{e',d'}$-subgraphs (that is, having size $e'$ and defect $d'$) in the high-girth $(r-2)$-graph ${\mathcal{K}}_i$ for some pairs $(e',d')$. Since the hypergraph ${\mathcal{K}}_i$ is linear, each drawn intersection has size 1. For $(e',d')$ in $\{(2,1),(3,1),(3,3)\}$, the family $\mathcal{S}_{e',d'}({\mathcal{K}}_i)$ consists of a unique $(r-2)$-graph up to isomorphism. For $(e',d')$ in $\{(3,2),(4,5)\}$, there are exactly two non-isomorphic examples. For the remaining pairs $(e',d')$, we provide a non-exhaustive list.
• Figure 2: The figure depicts the subgraph of the 3-graph $F_{63}$ "lying" on the pair $x_2x_3$. The central vertex in the figure is $x_1$, and the green diamonds correspond to $D_1$ and $D_1'$. Copies of the same construction "lie" on the pairs $x_1x_2$ and $x_2x_3$ in $F_{63}$.
• Figure 3: Configurations $P_3$ and $C_3$.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: BCD23*Theorem 3
• Theorem 1.6
• Lemma 3.1: sh*Lemma 5
• Theorem 4.1: GJKKLP*Theorem 3.1
• Lemma 4.2: mcdiarmid*Lemma 1.2
• Theorem 4.3: GJKKL*Theorem 1.4