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On the $(k+2,k)$-problem of Brown, Erdős and Sós for even integers $k$

Yan Wang, Jiasheng Zeng

Abstract

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $r$-graph on $n$ vertices in which every $k$ edges span more than $s$ vertices. Brown, Erdős and Sós in 1973 conjectured that for every $k\geq 2$, the limit $\lim_{n\to\infty} n^{-2} f^{(3)}(n;k+2,k)$ exists and verified the conjecture for $k=2$ by showing that $\lim_{n\to\infty} n^{-2} f^{(3)}(n;4,2)=\frac{1}{6}$. Delcourt and Postle, building on the work of Glock, Joos, Kim, Kühn, Lichev and Pikhurko, proved that for every $k\geq 2$, the limit $\lim_{n\to\infty} n^{-2} f^{(3)}(n;k+2,k)$ exists, thereby solving this conjecture. Their approach was later generalised by Shangguan to every uniformity $r\geq 4$: the limit $\lim_{n\to\infty} n^{-2} f^{(r)}(n; rk-2k+2,k)$ exists for all $r\geq 3$ and $k\geq 2$. However, its exact value was not determined. When $k\in\{2,3,\ldots,7\}$, the exact values of $\lim_{n\to\infty} n^{-2} f^{(r)}(n; rk-2k+2,k)$ were determined by Glock, Joos, Kim, Kühn, Lichev, Pikhurko, Rödl and Sun. Very recently, the limit for $k=8$ and $r\geq 4$ was determined by Pikhurko and Sun. For a general even integer $k$, Letzter and Sgueglia obtained the exact values of $\lim_{n\to\infty} n^{-2} f^{(r)}(n;rk-2k+2,k)$ for every even integer $k$ and uniformity $r\geq 2+\sqrt{2}\,k^{3/2}$. In this paper, we determine the exact value of $\lim_{n\to\infty} n^{-2} f^{(r)}(n;rk-2k+2,k)$ for every even integer $k\geq 4$ and $r\geq 2+\sqrt{\frac{3}{2}k-4}$, and show that it is $\frac{1}{r^2-r}.$

On the $(k+2,k)$-problem of Brown, Erdős and Sós for even integers $k$

Abstract

Let denote the maximum number of edges in an -graph on vertices in which every edges span more than vertices. Brown, Erdős and Sós in 1973 conjectured that for every , the limit exists and verified the conjecture for by showing that . Delcourt and Postle, building on the work of Glock, Joos, Kim, Kühn, Lichev and Pikhurko, proved that for every , the limit exists, thereby solving this conjecture. Their approach was later generalised by Shangguan to every uniformity : the limit exists for all and . However, its exact value was not determined. When , the exact values of were determined by Glock, Joos, Kim, Kühn, Lichev, Pikhurko, Rödl and Sun. Very recently, the limit for and was determined by Pikhurko and Sun. For a general even integer , Letzter and Sgueglia obtained the exact values of for every even integer and uniformity . In this paper, we determine the exact value of for every even integer and , and show that it is
Paper Structure (10 sections, 14 theorems, 56 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main large r even k sqrtk']}
  4. Lower bound
  5. Upper bound
  6. Mergeability
  7. Structural results for
  8. Assigning weights
  9. Concluding remarks
  10. Acknowledgment

Key Result

Theorem 1.1

For every even integer $k\geq 4$ and integer $r\geq \sqrt{2}k^{\frac{3}{2}}+2$, we have $\pi(r,k)=\frac{1}{r^2-r}$.

Figures (1)

  • Figure 3.1: This is an example of a $4$-uniform hypergraph with Property $\mathbb{P}$ in the case when $k=6$ and $r=4$. Each quadrilateral represents a hyperedge. If the green edges are taken as $T$, then the two red diamonds can serve as two flexible diamonds $T_1$ and $T_{\ell}$ in the definition.

Theorems & Definitions (25)

  • Theorem 1.1: letzter2025problemSIAM-k1.5
  • Theorem 1.2
  • Lemma 3.1: glock20246-k4,Theorem 3.1
  • proof : Proof of the lower bound of Theorem \ref{['main large r even k sqrtk']}
  • Lemma 3.2
  • Lemma 3.3: Trimming Lemma Glock_Kim_Lichev_Pikhurko_Sun_2025
  • Corollary 3.4
  • Lemma 3.5
  • Theorem 3.6
  • proof
  • ...and 15 more