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arxiv: 2604.11495 · v1 · submitted 2026-04-13 · 🧮 math.CO

Forbidding matching as trace in uniform hypergraphs

Yichen Wang , Xin Cheng , Ervin Gy\H{o}ri , Xiamiao Zhao This is my paper

Pith reviewed 2026-05-10 16:00 UTC · model grok-4.3

classification 🧮 math.CO
keywords extremal hypergraph theorytrace containmentmatchingTurán numbersuniform hypergraphsgeneralized Turán numberforbidden subhypergraphs
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The pith

The maximum number of edges in an r-uniform hypergraph on n vertices without containing a matching of s+1 edges as a trace is bounded by an expression that is asymptotically tight, with an exact determination when r=3.

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

This paper studies the extremal function for avoiding a matching as a trace in uniform hypergraphs. A hypergraph contains another as trace if some vertex subset induces the forbidden structure via edge intersections. The authors establish an upper bound on the number of edges possible without such a trace, showing it is asymptotically optimal by matching constructions. For three-uniform hypergraphs, they determine the exact maximum. They extend the result to the generalized Turán problem of maximizing the number of r-cliques under the same forbidden trace.

Core claim

The paper proves that ex_r(n, Tr_r(M_{s+1})) admits an upper bound that is sharp asymptotically as n tends to infinity. When r=3 the value is determined exactly. Analogous results hold for the maximum number of K_t^r copies in an n-vertex r-uniform hypergraph that does not contain Tr_r(M_{s+1}) as a trace, again with asymptotic sharpness and exactness for r=3.

What carries the argument

The trace operation, which extracts a hypergraph on a vertex subset S by taking all nonempty intersections of the original edges with S, and the extremal number ex_r(n, Tr_r(G)) measuring the maximum edges avoiding G as such a trace.

If this is right

  • The generalized Turán number for maximizing K_t^r copies while avoiding the matching trace obeys the same asymptotically sharp upper bound.
  • When r=3 both the ordinary and generalized extremal numbers are known exactly.
  • An analogue result holds for the mixed problem of simultaneously forbidding a matching trace and a complete-graph trace.

Where Pith is reading between the lines

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

  • Explicit constructions for general r could upgrade the asymptotic bounds to exact determinations.
  • The trace framework may let results on forbidden matchings transfer from graphs to hypergraphs more directly than ordinary subgraph containment.
  • Stability versions of the proofs would characterize the structure of hypergraphs attaining nearly the maximum number of edges.

Load-bearing premise

The asymptotic sharpness relies on the existence of suitable constructions achieving the upper bound, and the trace-containment definition holds without additional restrictions on degrees or intersections.

What would settle it

An r-uniform hypergraph on sufficiently large n with strictly more edges than the stated upper bound, yet with no vertex subset whose edge intersections contain a matching of s+1 edges, would falsify the bound.

Figures

Figures reproduced from arXiv: 2604.11495 by Ervin Gy\H{o}ri, Xiamiao Zhao, Xin Cheng, Yichen Wang.

Figure 1
Figure 1. Figure 1: The dotted lines are the missing edges in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

We say a hypergraph $\mathcal{H}$ contains a hypergraph $\mathcal{G}$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = |V(\mathcal{G})|$ and $\{e \cap S: e \in E(\mathcal{H})\}$ contains $\mathcal{G}$ as a sub-hypergraph. We use $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$ to denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$ vertices not containing $\mathcal{G}$ as a trace. The study of Tur\'{a}n numbers for traces was initiated by Mubayi and Zhao who studied the case when $\mathcal{G}$ is a complete graph. Let $M_{s+1}$ denote the graph of a matching with $s+1$ edges. In this paper, we give the upper bound of $\mathrm{ex}_r(n, \mathrm{Tr}_r(M_{s+1}))$ which is sharp asymptotically. When $r=3$, we give the exact value of $\mathrm{ex}_3 (n, \mathrm{Tr}_3 (M_{s+1}))$. We also consider the generalized Tur\'{a}n number in the case of matching. That is, the maximum number of copies of clique $\mathcal{K}_t^r$ in hypergraphs forbidding $\mathrm{Tr}_r (M_{s+1})$ as a trace. We give an upper bound which is sharp asymptotically and when $r=3$, we give the exact value. The Tur\'{a}n number of forbidding a matching and the other graph is another well studied topic initiated by Alon and Frankl. We also consider an analogue problem for the trace version, i.e., forbidding trace of matching and trace of complete graph as subgraphs.

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 paper defines the trace of an r-uniform hypergraph G in an r-uniform host H and studies the extremal function ex_r(n, Tr_r(M_{s+1})), where M_{s+1} is a matching of s+1 edges. It claims an asymptotically sharp upper bound on this function for general r, the exact value when r=3, and parallel results for the generalized Turán number ex(n, K_t^r, Tr_r(M_{s+1})). The work also briefly considers forbidding both a matching trace and a complete-graph trace simultaneously.

Significance. If the claimed bounds and exact determinations hold, the results extend the trace-Turán program initiated by Mubayi-Zhao from complete graphs to matchings, supplying both asymptotic sharpness and exact r=3 determinations that could serve as benchmarks for further trace problems. The generalized-Turán extension is a natural and potentially useful addition.

minor comments (3)
  1. The abstract and introduction state that the upper bound is 'sharp asymptotically' and that exact values are given for r=3, but the explicit form of the bound and the matching construction achieving it should be stated in the abstract or first paragraph of the introduction for immediate readability.
  2. Notation: the symbol Tr_r(G) is introduced without a displayed definition of the trace operation; a short displayed equation or diagram in §1 would clarify the intersection condition |S|=|V(G)| and the induced edge set.
  3. The final paragraph on the 'analogue problem for the trace version' of forbidding both matching and complete-graph traces is only sketched; either expand it into a short section with a precise statement or move it to a concluding remarks paragraph.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance in extending the trace-Turán program from complete graphs to matchings, and the recommendation for minor revision. The referee's summary accurately captures the main contributions: the asymptotically sharp upper bound on ex_r(n, Tr_r(M_{s+1})), the exact determination for r=3, and the parallel results for the generalized Turán number ex(n, K_t^r, Tr_r(M_{s+1})). Since the report contains no specific major comments or requests for changes, we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the trace extremal function ex_r(n, Tr_r(M_{s+1})) and supplies new upper bounds that are asymptotically sharp, together with exact values for r=3 and analogous results for the generalized Turán number ex(n, K_t^r, Tr_r(M_{s+1})). These claims rest on the given definition of trace containment and on explicit matching constructions; no equation or bound reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation. Background references to Mubayi-Zhao and Alon-Frankl supply historical context but are not invoked to justify the new asymptotic or exact statements. The derivation chain therefore remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard definitions of r-uniform hypergraphs and trace containment introduced by Mubayi and Zhao; no new free parameters, ad-hoc axioms, or invented entities appear in the abstract.

axioms (1)
  • standard math Standard definition of trace containment in hypergraphs
    Invoked throughout the abstract as the central notion.

pith-pipeline@v0.9.0 · 5656 in / 1265 out tokens · 43742 ms · 2026-05-10T16:00:04.169977+00:00 · methodology

discussion (0)

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Reference graph

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