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Lagrangians are attained as uniform Tur\'an densities

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Pith's one-line read Every palette Lagrangian is attained as a uniform Turán density, so the set of uniform Turán densities is not well-ordered.

desk verdict A real converse to Lamaison: palette Lagrangians are attained as uniform Turán densities; the upper-bound proof is solid, but the lower bound rests on an unproved folklore lemma. read the letter →

arxiv 2412.07297 v1 pith:P3F4R4TS submitted 2024-12-10 math.CO

Dylan King , Marcelo Sales , Bjarne Schülke This is my paper · ORCID
classification math.CO MSC 05C6505D0505D40
keywords uniformTurándensitypaletteLagrangianhypergraph3-uniformhypergraphsnon-well-ordereddensitiesquasirandomjumpingconjectureextremalcombinatorics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the palette Lagrangians—optimization numbers built from finite sets of ordered colour triples—are all attained as uniform Turán densities of families of 3-uniform hypergraphs. This transfers a known structural fact about ordinary Turán densities to the uniform setting: adding the scaled Lagrangians $(t/6)\Lambda^{(3)}$ for every $1 \le t \le 6$ into the set of attainable densities, where $\Lambda^{(3)}$ is the set of ordinary 3-graph Lagrangians. A direct consequence is that the set $\Pi^{(3)}_{\infty}$ of uniform Turán densities of families contains an infinite strictly decreasing sequence and is therefore not well-ordered. The proof works simultaneously for three variants of quasidense hypergraphs, each defined by a different way of measuring edge counts on pairs or triples of subsets.

What carries the argument

The engine is the palette Lagrangian. A palette $P$ is a finite set of ordered triples of colours; a 3-graph satisfies $P$ if its vertices can be ordered and its pairs coloured so that every edge's ordered colour triple lies in $P$. For a weighting $x$ of the colours, $\lambda_P(x)=\sum_{(a,b,c)\in P}x_ax_bx_c$, and the palette Lagrangian $\Lambda_P$ is the maximum over all weightings, with three variants that take minima over ordered degrees or codegrees before maximising. Theorem 5.1 is the load-bearing mechanism: it controls the density of any large quasidense 3-graph that almost satisfies $P$ by $\Lambda_P$ up to an error linear in the irregularity $\eta$. The proof obtains that control through the multicolour regularity lemma, a Ramsey-theoretic reduction to three clusters with a monochromatic triangle of density vectors, and the hypergraph counting lemma.

What would settle it

Take a small palette $P$, compute $\Lambda_P$ by optimizing over weightings, and search exhaustively at growing $n$ for a 3-graph satisfying $P$ whose edge density exceeds $\Lambda_P+\varepsilon$ for a fixed $\varepsilon>0$; such a sequence would violate Theorem 5.1 and hence Theorem 3.5.

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Extended reading notes

Core claim

The central claim is Theorem 3.5: for every palette $P$ and each of the three density notions, the palette Lagrangian $\Lambda_P$ is attained as $\pi(\mathcal F_P)$, where $\mathcal F_P$ is the family of all 3-graphs that do not satisfy $P$. The proof rests on Theorem 5.1, a stability bound: if a large 3-graph $\alpha$-almost satisfies $P$ (after deleting at most $\alpha|V|^3$ edges) and is $(d,\eta)$-dense in any of the three senses, then $d \le \Lambda_P + C\eta + \nu$ for constants $C$ depending only on $P$ and $\nu$. Lemma 3.4 provides the matching lower bound $\pi(\mathcal F_P) \ge \Lambda_P$, so the density of $\mathcal F_P$ is squeezed to $\Lambda_P$. From this the paper derives Theorem 1.2, that $\frac{t}{6}\Lambda^{(3)} \subseteq \Pi^{(3)}_{\infty}$ for $1 \le t \le 6$, and Theorem 1.1, that $\Pi^{(3)}_{\infty}$ is not well-ordered.

Load-bearing premise

The lower-bound half of the main theorem rests on the folklore claim (Lemma 3.4) that for every palette $P$ and every $\eta>0$ there exist arbitrarily large 3-graphs that satisfy $P$ and are $(\Lambda_P,\eta)$-dense; if that random construction fails in any of the three density notions, attainment collapses to an upper bound only.

Editorial extensions

If this is right

  • Every $3$-graph Lagrangian $\Lambda$ yields six uniform Turán densities $\frac{t}{6}\Lambda$ for $t=1,\dots,6$.
  • The set $\Pi^{(3)}_{\infty}$ of uniform Turán densities of families is not well-ordered.
  • Combined with the approximation result cited in the paper, palette Lagrangians and uniform Turán densities of families coincide: $\Pi^{(3)}_{\infty}=\Lambda^{\mathrm{pal}}$.
  • Exact new values enter the set: $1/27$, $1/16$, $1/25$, and $1/27$ from tight cycles, and $(5\sqrt{5}+63)/1922$ from the hypergraph $F_{3,2}$.
  • By the strong hypergraph removal lemma, every palette Lagrangian is approximable by uniform Turán densities of finite families, so the finite-family set is not well-ordered.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the approximation result cited as [18] extends to the other two density notions, then Theorem 3.5 would yield the exact equality $\Pi^{(3)}_{\bullet,\infty}=\Lambda^{\mathrm{pal}}$ for all three notions; the paper itself only records the inclusion proved here.
  • The proof identifies, for each palette $P$, an explicit extremal family (all 3-graphs not satisfying $P$); determining which of these families are finite would connect directly to the finite-family strengthening that the paper leaves open.
  • Because the argument scales ordinary Lagrangians by $t/6$ through the six orderings of a triple, a natural test is whether $k!$-scaled $k$-graph Lagrangians are attained for $k>3$; the palette machinery suggests an affirmative analogue.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper studies uniform Turán densities of 3-uniform hypergraphs. Its main theorem, Theorem 3.5, states that for each of the three quasirandom density notions ‹ ∈ {τ, π, σ}, every palette Lagrangian Λ^‹_P is attained as π_‹(F) for the family F of all 3-graphs that do not satisfy the palette P. The proof is a sandwich: the lower bound comes from Lemma 3.4, a probabilistic palette construction stated as folklore, and the upper bound comes from Theorem 5.1, a regularity-based argument showing that any 3-graph that almost satisfies P and is (d,η,‹)-dense has d ≤ Λ^‹_P + O(η) + ν. The authors derive two corollaries: for every 1≤t≤6, the scaled Lagrangian set (t/6)Λ^{(3)} is contained in Π^{(3)}_{τ,∞}, and consequently Π^{(3)}_{τ,∞} is not well-ordered. They also list several explicit Lagrangian values as new uniform Turán densities in Observation 6.1.

Significance. If the missing lower-bound construction is properly supplied, this is a strong and interesting paper. It transfers the classical relation Λ^{(3)} ⊆ Π^{(3)}_∞ = closure(Λ^{(3)}) to the uniform Turán density setting, gives a new proof of the analogue of the Frankl–Rödl non-well-ordering result, and produces explicit irrational and algebraic members of Π^{(3)}_{τ,∞}. The main technical engine, Theorem 5.1, is a substantial regularity argument; the authors carefully track error terms through the hierarchy in (5.1) and treat the three density variants separately. A notable strength is that the upper-bound proof is not circular: it relies only on standard tools such as the multicolour regularity lemma and the counting lemma, and no parameter is fitted to the conclusion. The principal weakness is that the lower-bound Lemma 3.4 is asserted without proof or citation, and that lemma is load-bearing for the equality in Theorem 3.5.

major comments (1)
  1. [§3, Lemma 3.4] The lower bound in Theorem 3.5 rests entirely on Lemma 3.4, which is stated as folklore and given no proof or citation. In the proof of Theorem 3.5, the inequality Λ^‹_P ≤ π_‹(F) is exactly this lemma. The random palette construction is routine for the τ density, but for the π and σ densities the uniformity requirements are stronger: one must control e_π(X,P) and e_σ(P,Q) for every vertex set X and every pair graph P,Q, and the target value is the minimum-type Lagrangian λ_P or λ_P, not the average λ_P. Please provide a full proof of Lemma 3.4, or a precise reference that covers all three density notions. Without this, Theorem 3.5 is only an upper-bound result.
minor comments (5)
  1. [§6, Observation 6.1] The six claimed Lagrangian values are asserted without proof. Since these values are presented as applications of Theorem 1.2, please include derivations or references, especially for the algebraic value (1/6)Λ_{F_{3,2}} = (5√5+63)/1922 and for the tight-cycle values for ℓ ≥ 6.
  2. [§5, Lemma 5.2] There is a typo in the proof: 'for for a∉A' should read 'for a∉A'.
  3. [§5, hierarchy (5.1)] The notation '!' in the parameter hierarchy is not defined. Please state explicitly that it means 'chosen sufficiently small relative to' or 'much smaller than'.
  4. [§5, Case 3] The sentence 'we may assume that the minimum in λ^{a,b}_P(x) is obtained when a and b are in the (X1,X2) and (X2,X3) positions' is terse. Because the σ-density e_σ(P,Q) has a specified orientation, please spell out how the other possible position pairs are handled by the same counting argument.
  5. [§4, proof of Theorem 1.1] The step from the approximating sequence π_n ∈ Π^{(3)}_∞ to a strictly decreasing sequence λ_n ∈ Λ^{(3)} with the same limit should be justified in one sentence; it is standard but not immediate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the attainment theorem is proved by independent lower and upper bounds, with no parameter fitted and no load-bearing self-citation.

full rationale

The derivation chain is not circular. Theorem 3.5 is proved by sandwiching Λ‹_P ≤ π‹(F) ≤ Λ‹_P + 3ν, where F is the family of all 3-graphs not satisfying P. The lower bound is Lemma 3.4, which is asserted as a folklore probabilistic palette construction, and the upper bound is Theorem 5.1, whose proof uses the multicolour regularity lemma, the counting lemma, and the compactness-based Lemma 5.2. These are standard external results; no parameter is fitted to the conclusion, and the extremal quantity π‹(F) is not defined in terms of Λ‹_P. Theorem 1.2 is a direct algebraic construction of a palette from a hypergraph with λ_P(x) = (t/6)λ_F(x), and Theorem 1.1 transfers a decreasing sequence from Π^{(3)}_8 to Π^{(3)}_{∴,∞} via known results by Frankl–Rödl, Brown–Simonovits, and Pikhurko. There are no author self-citations in the proof chain. The only notable weakness is that Lemma 3.4 is stated as folklore without proof or citation, so the lower-bound half of Theorem 3.5 is not fully demonstrated in the manuscript; however, this is an omitted justification, not a circular step, since Lemma 3.4 is logically independent of the theorem and does not assume the conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; all constants in (5.1) are universal choices in the proof. No new physical or mathematical entities are postulated; palettes are defined combinatorial objects. The only notable unproved input is the folklore lower-bound construction in Lemma 3.4.

assumptions (6)
  • standard math Multicolour Regularity Lemma (Theorem 2.5) holds with interval-refined partitions and ε-color-regular pairs.
    Invoked in Claim 5.3 to find three large regular sets X1, X2, X3 from the colored complete graph.
  • standard math Counting Lemma (Lemma 2.6) gives approximate counts of colored triangles in regular triples.
    Used in all three cases of Theorem 5.1 to bound sums of |T_abc| by products of color densities.
  • standard math The Turán density of the complete 3-graph K_{t1}^{(3)} is strictly less than 1.
    Used in the parameter choice of (5.1) to force a large clique in the auxiliary 3-graph R.
  • domain assumption Frankl-Rödl non-jump theorem: there is an α in (2/9, 1) and a strictly decreasing sequence in Π^{(3)}_8 converging to α.
    Used in the proof of Theorem 1.1 to obtain a decreasing sequence of Lagrangians via (1.2).
  • domain assumption Brown-Simonovits-Pikhurko identity: Λ^{(k)} ⊆ Π^{(k)}_8 = closure of Λ^{(k)}.
    Used in the proof of Theorem 1.1 to transfer the decreasing sequence from ordinary Turán densities to Lagrangians.
  • domain assumption Lemma 3.4: palettes yield lower-bound constructions for π_‹(F). The paper calls it folklore and gives no proof or citation.
    Provides the lower-bound half of Theorem 3.5; this is the least-supported input in the paper.

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Pith. "Pith review of Lagrangians are attained as uniform Tur\'an densities." pith.science (2026). https://pith.science/paper/P3F4R4TS

@misc{pith2026241207297,
  author       = {Pith},
  title        = {Pith review of: Lagrangians are attained as uniform Tur\'an densities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3F4R4TS}},
  note         = {Machine review of arXiv:2412.07297}
}
abstract

The study of uniform Tur\'an densities was initiated in the 1980s by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an density of $F$, $\pi_{\therefore}(F)$, is defined as the infimum $d\in[0,1]$ such that every $3$-graph $H$ in which every linearly sized $S\subseteq V(H)$ induces at least $(d+o(1))\binom{\vert S\vert}{3}$ edges must contain a copy of $F$. Disproving Erd\H{o}s's famous jumping conjecture, Frankl and R\"odl showed that the set of Tur\'an densities is not well-ordered. We prove an analogous result for the uniform Tur\'an density, namely that the set $\Pi^{(3)}_{\therefore,\infty}=\{\pi_{\therefore}(\mathcal{F}) : \mathcal{F}\text{ a family of }3\text{-graphs} \}$ is not well-ordered. This is a consequence of a more general result, which in particular implies that for every Lagrangian $\Lambda$ of a $3$-graph and integer $1 \leq t \leq 6$ we have $\frac{t}{6}\Lambda\in \Pi^{(3)}_{\therefore,\infty}$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Intervals of uniform Tur\'an densities

    math.CO 2026-08 accept novelty 8.0 of 10

    For 3-uniform hypergraphs, every density in an interval ending at 1 is exactly the uniform Turán density of some possibly infinite forbidden family.

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