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arxiv: 2408.09029 · v2 · pith:KTWJPZRUnew · submitted 2024-08-16 · 🧮 math.CO

An Improved Tur\'an Exponent for 2-Complexes

Maya Sankar This is my paper

Pith reviewed 2026-05-23 22:11 UTC · model grok-4.3

classification 🧮 math.CO
keywords topological Turán number2-complexes3-uniform hypergraphsextremal combinatoricsTurán exponent4-cyclesdisk triangulationssimplicial complexes
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The pith

Any 2-complex has topological Turán number at most order n to the 8/3.

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

The paper proves that for any fixed 2-dimensional simplicial complex X the maximum number of edges in an n-vertex 3-uniform hypergraph containing no triangulation of X is at most C n to the power 8/3. This replaces an earlier uniform upper bound whose exponent was 3 minus 1/5. A reader interested in extremal hypergraph problems with topological constraints would see the result as a uniform improvement that applies to every such X rather than to special cases only. The argument also supplies streamlined upper bounds for the torus and real projective plane that extend to all surfaces.

Core claim

We prove that the Turán exponent of any such space X is at most 8/3, i.e., that ex_hom(n,X) is at most C n^{8/3} for some constant C depending only on X. The proof rests on a refined count of how 4-cycles that bound a disk triangulation sit inside a randomly chosen vertex subset, which improves the deletion or counting step that previously gave the weaker exponent 3-1/5. The same counting technique yields asymptotically tight bounds for the torus and the real projective plane and therefore for every surface.

What carries the argument

The distribution of 4-cycles vwv'w' that bound a disk triangulation inside a randomly selected vertex subset.

If this is right

  • The n^{8/3} bound holds for every 2-complex, not merely for surfaces or other special cases.
  • New proofs give asymptotically tight upper bounds for the torus and real projective plane.
  • Asymptotically tight upper bounds follow for the topological Turán numbers of all surfaces.
  • The earlier uniform exponent 3-1/5 is replaced by the strictly smaller 8/3.

Where Pith is reading between the lines

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

  • The same subset-sampling analysis might tighten bounds for other forbidden configurations that involve bounded disks.
  • If 8/3 turns out to be tight for some X, the random-subset distribution would be the natural obstacle preventing a better uniform exponent.
  • Direct computation for small fixed X and moderate n could give evidence on the size of the constant C.

Load-bearing premise

The placement of those 4-cycles inside random subsets obeys a regularity that lets the counting argument reach exponent 8/3.

What would settle it

A single 2-complex X together with a family of 3-uniform hypergraphs on n vertices that have more than n^{8/3+epsilon} edges and still contain no triangulation of X would disprove the claimed bound.

Figures

Figures reproduced from arXiv: 2408.09029 by Maya Sankar.

Figure 1
Figure 1. Figure 1: Part of Γ4 As another application of our analysis of disk-coverable cycles, we derive shorter proofs of the upper bounds for the topological Tur´an numbers of the torus T 2 and real projective plane RP2 . These two bounds are the main results of [14] and [20], and can be leveraged to prove an exponent of 5/2 for all orientable and nonorientable surfaces, respectively. Theorem 1.3 ([14] and [20]). We have e… view at source ↗
Figure 2
Figure 2. Figure 2: Decompositions of (i) the torus and (ii) the real projective plane as CW complexes. We construct homeomorphs of T 2 and RP2 by using Lemma 3.1 to attach disks to those (p, ǫ)-disk-coverable 4-cycles pictured in [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

The topological Tur\'an number $\mathrm{ex}_{\hom}(n,X)$ of a 2-dimensional simplicial complex $X$ asks for the maximum number of edges in an $n$-vertex 3-uniform hypergraph containing no triangulation of $X$ as a subgraph. We prove that the Tur\'an exponent of any such space $X$ is at most $8/3$, i.e., that $\mathrm{ex}_{\hom}(n,X)\leq Cn^{8/3}$ for some constant $C=C(X)$. This improves on the previous exponent of $3-1/5$, due to Keevash, Long, Narayanan, and Scott. Additionally, we present new streamlined proofs of the asymptotically tight upper bounds for the topological Tur\'an numbers of the torus and real projective plane, which can be used to derive asymptotically tight upper bounds for all surfaces. The key insight is an improved understanding of the placement of 4-cycles $vwv'w'$ that are likely to bound a triangulation of the disk within a randomly-selected subset of vertices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for any fixed 2-dimensional simplicial complex X the topological Turán number satisfies ex_hom(n,X) ≤ C n^{8/3} with C = C(X). This improves the prior general exponent 14/5 of Keevash-Long-Narayanan-Scott. The authors also supply streamlined proofs of the asymptotically tight upper bounds for the torus and real projective plane (hence for all surfaces). The argument proceeds by a random-subset deletion method whose improvement rests on a refined counting argument for the distribution of 4-cycles vwv'w' that bound a disk triangulation inside a random vertex subset.

Significance. If the claimed counting argument holds, the work supplies a concrete, parameter-free improvement of the general upper bound in the topological Turán problem for 3-uniform hypergraphs. The explicit distribution estimate for disk-bounding 4-cycles is a technical strength that directly yields the 8/3 exponent and recovers the earlier 14/5 bound as a special case when the estimate is weakened. The new surface proofs are additionally useful for applications.

minor comments (3)
  1. The abstract states that the key insight is an improved understanding of 4-cycle placement, but the introduction does not contain a one-paragraph roadmap that isolates the new counting lemma from the standard deletion framework; adding such a paragraph would help readers locate the improvement.
  2. Notation: the symbol ex_hom is introduced in the abstract without a parenthetical gloss; a brief definition on first use would improve accessibility for readers outside the immediate subfield.
  3. The dependence of the constant C on X is stated but not quantified; a short remark on how the proof tracks the dependence (e.g., via the number of simplices in X) would be helpful even if the focus is the exponent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of the 8/3 exponent improvement, and the recommendation for minor revision. We appreciate the note on the technical strength of the 4-cycle distribution estimate.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes an upper bound ex_hom(n,X) ≤ C n^{8/3} via an explicit random-subset counting argument on the distribution of disk-bounding 4-cycles inside vertex subsets; this counting step directly yields the improved exponent without reducing to a fitted parameter, a self-definition, or a load-bearing self-citation. The prior 14/5 exponent is recovered as a weakening of the same argument, confirming the improvement is localized and independent. Streamlined proofs for the torus and RP2 are presented as self-contained derivations that do not invoke the authors' prior uniqueness results or ansatzes. No equations or reductions in the provided text equate the claimed bound to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a combinatorial proof relying on standard counting and probabilistic arguments in extremal graph theory; no free parameters, ad-hoc axioms, or invented entities are indicated.

pith-pipeline@v0.9.0 · 5710 in / 969 out tokens · 22371 ms · 2026-05-23T22:11:36.288445+00:00 · methodology

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Reference graph

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