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arxiv: 2411.01782 · v3 · submitted 2024-11-04 · 🧮 math.CO

The Tur\'an Density of 4-Uniform Tight Cycles

Maya Sankar This is my paper

Pith reviewed 2026-05-23 18:12 UTC · model grok-4.3

classification 🧮 math.CO
keywords Turán densitytight cycles4-uniform hypergraphshomomorphic avoidancecycle lengths modulo r
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The pith

Sufficiently long 4-uniform tight cycles whose length is not divisible by 4 have Turán density exactly 1/2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper provides an exact characterization of r-uniform hypergraphs that avoid homomorphic images of tight cycles of certain lengths modulo r, described via colorings of (r-1)-tuples. It outlines a general method showing that in families of tight-cycle-like structures where this characterization holds, all sufficiently long members share the same Turán density. Applying the method to 4-uniform tight cycles yields that beyond a certain length, those with length not a multiple of 4 have Turán density 1/2. This matters because it resolves the asymptotic maximum density of 4-uniform hypergraphs without these forbidden cycles for most lengths.

Core claim

There exists an integer L0 such that for every L > L0 not divisible by 4, the 4-uniform tight cycle of length L has Turán density 1/2. This follows from a general strategy that equates the Turán densities of long members in families of tight-cycle-like hypergraphs to which a coloring-based characterization of avoidance applies.

What carries the argument

The characterization of hypergraphs homomorphically avoiding tight cycles of length k modulo r by means of colorings of (r-1)-tuples of vertices, which generalizes bipartiteness.

If this is right

  • The Turán density is the same for all sufficiently long 4-uniform tight cycles not divisible by 4.
  • The characterization holds for a broad class of families beyond just the modulo-r cycle families.
  • All long enough members of applicable tight-cycle families share a common Turán density.
  • The result for uniformity 4 extends previous work on graphs and 3-uniform hypergraphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This framework could potentially determine Turán densities for other lengths or uniformities if the strategy applies similarly.
  • Connections to homomorphism properties might link to other extremal problems in hypergraph theory.
  • Testing the density for specific large L could confirm the limit behavior.

Load-bearing premise

The general strategy outlined for proving equal Turán densities in tight-cycle-like families applies specifically to 4-uniform tight cycles.

What would settle it

A construction of 4-uniform hypergraphs with asymptotic density strictly greater than 1/2 containing no tight cycle of some large length L not divisible by 4 would falsify the claim.

Figures

Figures reproduced from arXiv: 2411.01782 by Maya Sankar.

Figure 1
Figure 1. Figure 1: At left, a 3-graph G. At right, an (accordant) oriented coloring of E(G) by ∆ = {∆refl, ∆rot} [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: At left, an accordant oriented coloring of the edges of a bipartite graph by { }. At right, the corresponding 2-coloring of the vertices. Proposition 3.10. Fix an integer r and two residues s, k modulo r. Any C (r) sk -hom-free r-graph is also C (r) k -hom-free. Proof. Suppose G is C (r) ak -hom-free. For each ℓ ≡ k (mod r), it holds that G is C (r) sℓ -hom-free, and Proposition 2.3 implies that G is C (r)… view at source ↗
Figure 4
Figure 4. Figure 4: The three colorings of the four sub-triples of a 4-edge permitted by Corol￾lary 3.14(4). Proof. We have (1) ⇐⇒ (2) by either Proposition 3.10 or Remark 3.11. To show (1) ⇐⇒ (3), we apply Theorem 3.2 with r = 4 and k = 1. Let π = cyc1 = (1 2 3 4). There are 11 conjugacy classes of subgroups of S4; a complete list of these subgroups is given in Appendix A. The subgroups avoiding π are the conjugates of S3, A… view at source ↗
Figure 5
Figure 5. Figure 5: At left, a coloring of (V1 ∪ V2) (3) that proves Godd(V1, V2) is C (4) k -hom-free. At right, the corresponding accordant oriented coloring of E(Godd(V1, V2)). Proposition 3.15. Given any two disjoint sets of vertices V1 and V2, the complete oddly bipartite 4-graph Godd(V1, V2) is C (4) k -hom-free for k = 1, 2, 3. In particular, ex(n, C(4) L ) ≥ e opt(n) for each n and each L ̸≡ 0 (mod 4). Proof. Let V = … view at source ↗
Figure 6
Figure 6. Figure 6: The coloring of E(Kn) by { , } that maximizes the number of cherries. Xi+1 (mod 3). Each purple directed triangle contributes 3 such orientations, so H(X1, X2, X3) = log2  3T( )  . Moreover, (X1, X2) is supported on the set of purple edges (x, y) directed from x to y, which is a set of size 2δ [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: At left, the coloring of (V1 ∪ V2) (3) from [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: The number of edges colored with at least one endpoint in V2 is at most X v∈V2 bv ≤ |V ′ 2 | × ϵ2n + |Vbad| × n ≤ ϵ2n 2 + ϵ2n 2 = 2ϵ2n 2 . Thus, using (5.10.3), there are least β1 [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The conjectural extremal constructions for tight cycles of length relatively prime to the uniformity in uniformities 5 and 6. 7.2 Other families of cycle-like hypergraphs Let us mention two other families of cycle-like hypergraphs whose Tur´an densities might possibly be approached with our framework. First, we suspect that tight cycles in higher uniformities lie within reach. Here, the challenge is unders… view at source ↗
read the original abstract

For any uniformity $r$ and residue $k$ modulo $r$, we give an exact characterization of the $r$-uniform hypergraphs that homomorphically avoid tight cycles of length $k$ modulo $r$, in terms of colorings of $(r-1)$-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, as well as a result of Kam\v cev, Letzter, and Pokrovskiy in uniformity 3. In fact, our characterization applies to a much larger class of families than those of the form $\mathscr C_k^{(r)}=\{\text{$r$-uniform tight cycles of length $k$ modulo $r$}\}$. We also outline a general strategy to prove that, if $\mathscr C$ is a family of tight-cycle-like hypergraphs (including but not limited to the families $\mathscr C_k^{(r)}$) for which the above characterization applies, then all sufficiently long $C\in \mathscr C$ will have the same Tur\'an density. We demonstrate an application of this framework, proving that there exists an integer $L_0$ such that for every $L>L_0$ not divisible by 4, the tight cycle $C^{(4)}_L$ has Tur\'an density $1/2$.

Editorial analysis

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Referee Report

0 major / 3 minor

Summary. The paper provides an exact characterization of the r-uniform hypergraphs that homomorphically avoid tight cycles of length k modulo r, expressed in terms of colorings of (r-1)-tuples of vertices. This generalizes the characterization of bipartite graphs (avoiding odd closed walks) and a prior 3-uniform result of Kamčev, Letzter, and Pokrovskiy. The characterization is stated to apply to a broader class of families than just the tight cycles C_k^{(r)}. The authors outline a general strategy showing that, for families of tight-cycle-like hypergraphs to which the characterization applies, all sufficiently long members have identical Turán densities; they apply the strategy to prove that there exists L_0 such that every 4-uniform tight cycle C_L^{(4)} with L > L_0 not divisible by 4 has Turán density exactly 1/2.

Significance. If the proofs are complete, the characterization supplies a structural tool analogous to bipartiteness that could be applied to other homomorphism problems in hypergraph theory. The outlined strategy for equal Turán densities across long cycles is a methodological contribution that may extend beyond the specific 4-uniform case. The concrete result that the Turán density is 1/2 for all sufficiently long C_L^{(4)} (L not divisible by 4) is a notable exact determination in an area where hypergraph Turán densities are typically only bounded.

minor comments (3)
  1. [Abstract] The abstract introduces the notationmathscr C_k^{(r)} and the phrase 'tight-cycle-like hypergraphs' without a self-contained definition; ensure both are defined explicitly in §1 or §2 before the characterization is stated.
  2. [Abstract] The claim that the strategy applies to 'a much larger class of families' is stated without an example outside C_k^{(r)}; adding one concrete additional family in the introduction would clarify the scope.
  3. The existence of L_0 is asserted but no explicit bound or dependence on the characterization is indicated; if the proof yields a computable L_0, recording it would strengthen the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its significance, and recommendation for minor revision. The summary accurately captures the main contributions.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states a new characterization of r-uniform hypergraphs that homomorphically avoid tight cycles of length k mod r, expressed via colorings of (r-1)-tuples; this is presented as a generalization of known bipartite and 3-uniform results from independent prior work. It then outlines a separate general strategy for showing equal Turán densities among sufficiently long members of applicable families and applies the strategy to obtain the 1/2 density for long C_L^{(4)}. No quoted step reduces a claimed prediction or density result to a fitted parameter, self-definition, or load-bearing self-citation; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on a new characterization theorem for homomorphic avoidance and a general strategy for showing density equality across long cycles in the family; no free parameters or invented entities are mentioned.

axioms (1)
  • standard math Standard properties of hypergraph homomorphisms and colorings of (r-1)-tuples determine avoidance of tight cycles of length k mod r
    Invoked to obtain the exact characterization (abstract).

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Reference graph

Works this paper leans on

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    (Because f is continuous and R is closed and bounded, we know that f attains a maximum on R.) The proof is given in a sequence of propositions, each deriving further restrictions on the global maxima of f. Throughout, we 47 implicitly use that the value of f at its maximum ( α, β, γ, δ) on R must satisfy f(α, β, γ, δ) ≥ f 1 4 , 1 4 , 1 4 , 0 = 0.5. We fir...

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    Set f1(α, β, γ, δ) = γ3/2 +2δ3/2 +3α√β and f2(α, β, γ, δ) = γ3/2 +2δ3/2 + 3αβ α+β , so f0 = min(f1, f2)

    The next four propositions study the global maxima of f0 on R0. Set f1(α, β, γ, δ) = γ3/2 +2δ3/2 +3α√β and f2(α, β, γ, δ) = γ3/2 +2δ3/2 + 3αβ α+β , so f0 = min(f1, f2). 48 Proposition B.3. Let (α, β, γ, δ) be a global maximum of f0 on the region R0 defined above. Then either γ = 0 or γ = α or δ = 0 or δ = 0.18. Proof. Suppose, for the sake of contradictio...

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    Thus, f0(α, β, γ, δ) ≤ f1(α, β, γ, δ) = 2δ3/2 + α3/2 + 3α √ 1 − 3α = 2δ3/2 + j(α) ≤ 2δ3/2 + 1 2(1 − 2δ)3/2

    = (c/4)3/2 + 3(c/4) p c/4 = c3/2/2 = (1 − 2δ)3/2/2 on the interval 0, c 3 . Thus, f0(α, β, γ, δ) ≤ f1(α, β, γ, δ) = 2δ3/2 + α3/2 + 3α √ 1 − 3α = 2δ3/2 + j(α) ≤ 2δ3/2 + 1 2(1 − 2δ)3/2. 49 Observing that δ1/2 ≤ 0.181/2 < 1 2 and (1 − 2δ)1/2 ≤ 1, we have that f0(α, β, γ, δ) ≤ 2δ3/2 + 1 2(1 − 2δ)3/2 ≤ 1 2 × 2δ + 1 2 × (1 − 2δ) = 1 2 , with strict inequality i...

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    However, f0(α, β, γ, δ) = h1( 1 5 , 1 5 , 1

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  30. [30]

    ≈ 0.447 < 0.5, so this point is not a global maximum of f0 on R0. If δ = 0 then ( α, β, γ) must be a global maximum of j1(α, β, γ) := f1(α, β, γ, 0) on the region R′ 1 = {(α, β, γ) ∈ R3 : 2α + β + γ = 1, α > 0, β > 0, γ > 0.1, γ < α, p β < α + β} which is an open subset of the hyperplane {(α, β, γ) : 2α + β + γ = 1}. Because j1 is differentiable on R′ 1, ...

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  31. [31]

    It follows that f0(α, β, γ, δ) ≤ 1 2 with equality only if β = 1 4, in which case α = √β − β = 1 4 and γ = 1 − 2α − β = 1 4. Case 2. γ = 0. In this case, 2 δ = 1 − 2α − β = 1 − 2√β + 2β − β = (1 − √β)2, so f0(α, β, γ, δ) = 2δ3/2 + 3β − 3β3/2 = 1√ 2 1 − p β 3 + 3β 1 − p β ≤ 3 4 1 − p β 3 + 3β 1 − p β = 3 4 16 25 + 9 25 − 3 p β + 7β − 5β3/2 = 0.48 + 3 4 1 −...

    work page