Near-optimal Turán densities of $r$-graphs on $r+1$ vertices

Abstract: Let $π(H)$ be the Turán density of an r-uniform hypergraph $H$ and let $H_k<sup>r$ denote the $r$-uniform hypergraph on $r+1$ vertices with exactly $k$ edges, where $1\le k\le r+1$. Sidorenko~(JCT-B, 2024) proved that $π(H_3^r)\ge (1.7215-o(1))r<sup>{-2}$ as $r\to\infty$ and $$π(H_k^r)\ge (C_k+o(1))r<sup>{-(1+1/(k-2))}$$ for fixed $k$ as $r\to\infty$. Clemen~later improved the first bound to $$π(H_3^r)\ge cr^{{-2}\sqrt{\log} r}$$ for some constant $c&gt;0$. In this article, we prove the following results. \begin{itemize} \item For any fixed $\varepsilon&gt;0$, there is a constant $c_\varepsilon&gt;0$ such that $$π(H_3^r)\ge \frac{c_\varepsilon}{r(\log r)<sup>{2+\varepsilon}}.$$ %$π(H_3^r)\ge 1/(r(\log r)<sup>{2+o(1)})$. Together with the known upper bound $π(H_3<sup>r)\le1/r$, this implies $π(H_3<sup>r)=r<sup>{-1+o(1)}$. \item For every $3\le k\le r+1$, let $s=\min{k-2,r-k+2}$. Then \begin{equation*} 0\le \frac{k-2}{r}-π(H_k^r) \le \frac{128}{r}\left(\sqrt{s\log\frac{er}{s}}+\log\frac{er}{s}\right). \end{equation*} This estimate yields several asymptotically sharp results for $π(H_k<sup>r)$. For example, $π(H_k<sup>r)=(1+o(1))(k-2)/r$ when $\log(er/(k))=o(k)$. \end{itemize}

• The paper proves that π(H₃ʳ)=r⁻¹⁺ᵒ⁽¹⁾, improving earlier lower bounds through recursive blow-ups, local colourings, and pair-covering methods.
• For every 3≤k≤r+1, it shows that π(Hₖʳ) differs from the upper bound (k−2)/r by at most 128r⁻¹(√(s log(er/s))+log(er/s)), where s=min{k−2,r−k+2}.
• The circular construction and cyclic scan estimates become asymptotically sharp when k grows sufficiently with r, while the correct logarithmic factor for H₃ʳ and fixed-k densities remain open problems.

The paper under review, by Jiabao Yang and Xiutao Zhu (2608.18924), determines the asymptotic order of the Turán density $\pi(H_3^r)$ and gives a near-optimal finite estimate for $\pi(H_k^r)$ for all $3\le k\le r+1$, where $H_k^r$ denotes the unique (up to isomorphism) $r$-graph on $r+1$ vertices with exactly $k$ edges. The problem is equivalent to the Brown–Erdős–Sós function $f^{(r)}(n,r+1,k)$, since $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$. Prior to this work, the best lower bounds for $\pi(H_3^r)$ were $\pi(H_k^r)$0 (Sidorenko) and $\pi(H_k^r)$1 (Clemen), against an upper bound of $\pi(H_k^r)$2; for general $\pi(H_k^r)$3, Sidorenko's circular construction gave lower bounds of order $\pi(H_k^r)$4 for fixed $\pi(H_k^r)$5, far below the upper bound $\pi(H_k^r)$6.

Background and main results

For an $\pi(H_k^r)$7-uniform hypergraph $\pi(H_k^r)$8, $\pi(H_k^r)$9. Determining Turán densities of $3\le k\le r+1$0-graphs is notoriously difficult; even $3\le k\le r+1$1 remains open. The family $3\le k\le r+1$2 interpolates between classical cases: $3\le k\le r+1$3 is a triangle ($3\le k\le r+1$4 by Mantel), $3\le k\le r+1$5 (conjectured $3\le k\le r+1$6 by Frankl and Füredi), and $3\le k\le r+1$7. A useful monotonicity, $3\le k\le r+1$8, follows from considering link graphs.

The paper proves two main theorems. First, for every fixed $3\le k\le r+1$9 there exists $H_k^r$0 such that

$H_k^r$1

for all sufficiently large $H_k^r$2. Combined with the known upper bound $H_k^r$3, this yields the sharp-in-exponent conclusion

$H_k^r$4

i.e., $H_k^r$5. This resolves the exponent of $H_k^r$6 in the density, leaving only polylogarithmic factors undetermined.

Second, writing $H_k^r$7, the authors prove that for all integers $H_k^r$8 and $H_k^r$9,

$r$0

This closes the gap to the trivial upper bound $r$1 whenever $r$2 grows with $r$3. Three corollaries quantify this: if $r$4 then $r$5; if $r$6 then $r$7 exactly; and if $r$8 with $r$9, the deficit from $r+1$0 is at most $r+1$1.

Proof for $r+1$2: recursive blow-up via local colourings

The engine of the first result is a recursion: there is an absolute constant $r+1$3 such that for all $r+1$4,

$r+1$5

Iterating this map $r+1$6 reaches a fixed base value in $r+1$7 steps, and a product lemma shows the accumulated factor satisfies $r+1$8, giving the theorem.

The recursion itself is established through a two-coordinate labelling argument. The vertex set of size $r+1$9 is partitioned into $k$0 parts of size $k$1. A key ingredient is a colouring lemma: using independent random permutations of a dense $k$2-free family on each part, one can colour all $k$3-subsets of a part with at most $k$4 colours so that no $k$5-set contains three same-coloured $k$6-subsets. An $k$7-set meeting every part in at most $k$8 vertices is called good; a hypergeometric tail bound shows good sets constitute a $k$9 fraction of all $f^{(r)}(n,r+1,k)$0-sets. Each good set receives a label consisting of (a weighted sum of part indices mod $f^{(r)}(n,r+1,k)$1) and (a sum of local colours mod $f^{(r)}(n,r+1,k)$2). If three edges of a hypothetical copy of $f^{(r)}(n,r+1,k)$3 shared a label, the first coordinate forces the three deleted vertices into the same part, while the second coordinate contradicts the local colouring property. Taking the largest label class and applying Sidorenko's pair-covering blow-up lemma ($f^{(r)}(n,r+1,k)$4 for pair-covering $f^{(r)}(n,r+1,k)$5) yields the recursion.

An honest caveat: the constants are chosen so that the error term in the product lemma absorbs both the geometric decay of the sequence and the constant $f^{(r)}(n,r+1,k)$6; the result is therefore sensitive to these constants but robust in its logarithmic structure.

Proof for general $f^{(r)}(n,r+1,k)$7: circular construction and cyclic scan statistics

The second main theorem refines Sidorenko's circular construction. In that construction, $f^{(r)}(n,r+1,k)$8 points are placed uniformly at random on a circle of circumference one, and an $f^{(r)}(n,r+1,k)$9-set is an edge when the argument of the product of its points lies within the shortest arc containing $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$0 points; the resulting infinite $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$1-graph is $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$2-free with edge density $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$3, where $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$4 denotes the length of the shortest arc containing $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$5 points.

Writing the gaps as normalized spacings, Rényi's representation identifies $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$6 with $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$7 for i.i.d. exponential variables $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$8, independent of $\mathrm{ex}(n,H_k^r)=f^{(r)}(n,r+1,k)-1$9. Consequently $\pi(H_3^r)$0, where $\pi(H_3^r)$1 is the minimum over $\pi(H_3^r)$2 of the cyclic sum of $\pi(H_3^r)$3 consecutive exponentials. Centering via $\pi(H_3^r)$4, the problem reduces to bounding $\pi(H_3^r)$5 for cyclic sums of centered variables.

The key probabilistic estimate is a block decomposition: starting indices are grouped into blocks of size at most $\pi(H_3^r)$6, and within each block all relevant cyclic sums involve at most $\pi(H_3^r)$7 distinct consecutive variables, so a maximal Bernstein inequality for ordinary partial sums applies. A union bound over blocks and integration of the resulting tail gives

$\pi(H_3^r)$8

for $\pi(H_3^r)$9. For $\pi(H_k^r)$00, the complement identity $\pi(H_k^r)$01 with $\pi(H_k^r)$02 reduces the case to cyclic sums of length $\pi(H_k^r)$03. Combining both regimes with the known upper bound $\pi(H_k^r)$04 yields the finite estimate. The corollaries then follow by direct asymptotic analysis of $\pi(H_k^r)$05 under the three growth regimes for $\pi(H_k^r)$06.

It should be noted that for small fixed $\pi(H_k^r)$07 the error term exceeds $\pi(H_k^r)$08 itself — for $\pi(H_k^r)$09 it is much larger — so the estimate is informative precisely when $\pi(H_k^r)$10 grows with $\pi(H_k^r)$11; the fixed-$\pi(H_k^r)$12 regime retains the gap between $\pi(H_k^r)$13 and $\pi(H_k^r)$14.

Limitations and open questions

Several limitations are explicit. The bound $\pi(H_k^r)$15 does not determine whether the true density is closer to $\pi(H_k^r)$16 or carries a genuine logarithmic penalty; pinning down the correct power of $\pi(H_k^r)$17 (or showing none) remains open. The Frankl–Füredi conjecture $\pi(H_k^r)$18 and the determination of $\pi(H_k^r)$19 remain untouched. For fixed $\pi(H_k^r)$20, the multiplicative gap between the circular-construction lower bounds and the upper bound $\pi(H_k^r)$21 persists. Finally, the additive constant $\pi(H_k^r)$22 in the main estimate is not optimized, and the corollary in the regime $\pi(H_k^r)$23 leaves a non-vanishing relative gap of order $\pi(H_k^r)$24.

Conclusion

The paper establishes that $\pi(H_k^r)$25, upgrading the previously known $\pi(H_k^r)$26-type lower bounds by a recursive local-colouring and blow-up argument, and proves that $\pi(H_k^r)$27 differs from its trivial upper bound $\pi(H_k^r)$28 by at most $\pi(H_k^r)$29, which is asymptotically sharp whenever $\pi(H_k^r)$30. These results substantially narrow the Brown–Erdős–Sós-type extremal landscape for $\pi(H_k^r)$31-graphs on $\pi(H_k^r)$32 vertices, while leaving the precise polylogarithmic behaviour of $\pi(H_k^r)$33 and the fixed-$\pi(H_k^r)$34 regime as open problems.