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arxiv: 2604.17038 · v1 · submitted 2026-04-18 · 🧮 math.CO

Hypergraphs without Subgraphs of Given Connectivity

Jie Ma , Shengjie Xie , Zhiheng Zheng This is my paper

Pith reviewed 2026-05-10 06:38 UTC · model grok-4.3

classification 🧮 math.CO
keywords hypergraphsr-uniform hypergraphsconnectivityextremal problemssubgraph avoidanceseparator treesedge maxima
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The pith

The maximum number of edges in an n-vertex r-uniform hypergraph without a (k+1)-connected subgraph is determined up to an O(n) term for r at least 3.

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

This paper determines the maximum number of edges in an n-vertex r-uniform hypergraph with no (k+1)-connected subgraph. For all r at least 3, the value is found to within an additive error of O(n), which identifies the leading asymptotic term. It also gives a tight bound on edges for hypergraphs where no (k+1)-connected subgraph exceeds Ck vertices when C greater than 2 and r is large enough, along with an asymptotically tight bound for C equal to 2. The results extend Mader's classical problem from graphs to hypergraphs and resolve a question of Carmesin using separator tree methods augmented by new techniques.

Core claim

We study the problem of determining the maximum number of edges in an n-vertex r-uniform hypergraph that contains no (k+1)-connected subgraph. For all r greater than or equal to 3, we determine this maximum up to an O(n) error term, thereby identifying its leading term. We also address a related question of Carmesin by establishing a tight bound for r-uniform hypergraphs with no (k+1)-connected subgraph on more than Ck vertices for any constant C greater than 2 and sufficiently large r, and further obtain an asymptotically tight bound in the case C equals 2. Our proof combines the separator tree method with several new combinatorial and optimization techniques.

What carries the argument

Separator tree method of Carmesin combined with new combinatorial and optimization techniques that control edge counts while forbidding (k+1)-connected subgraphs in r-uniform hypergraphs.

If this is right

  • The leading term of the extremal function forbidding (k+1)-connected subgraphs is identified for all r-uniform hypergraphs with r at least 3.
  • A tight upper bound holds when every (k+1)-connected subgraph is restricted to at most Ck vertices for C greater than 2 and sufficiently large r.
  • An asymptotically tight bound is obtained in the case where C equals 2.
  • Related open problems on hypergraph connectivity and edge maxima are posed for further work.

Where Pith is reading between the lines

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

  • The hypergraph result may supply partial insight or a template for attacking the still-open graph version of the same extremal problem.
  • The bounds allow construction of high-edge-density hypergraphs whose connected components remain small.
  • The separator-tree approach could adapt to other forbidden-structure problems in uniform hypergraphs.

Load-bearing premise

The separator tree method extends without gaps from graphs to r-uniform hypergraphs for r at least 3 when combined with the new combinatorial and optimization techniques.

What would settle it

An r-uniform hypergraph on n vertices with no (k+1)-connected subgraph but containing more than the identified leading term plus some constant times n edges would show the claimed maximum is incorrect.

Figures

Figures reproduced from arXiv: 2604.17038 by Jie Ma, Shengjie Xie, Zhiheng Zheng.

Figure 1
Figure 1. Figure 1: On the left, we sketch an r-uniform hypergraph H recursively separated by the separators S1 and S2, where S1 = A ∩ (B ∪ C) and S2 = B ∩ C. The corresponding separator tree is shown on the right. The three atoms are labelled A, B, and C, each of size at most ck + k. In the remaining of this section, we fix an H ∈ Fc,r(n, k) and a separator tree TH of H. Given the separator tree, we obtain an inductive relat… view at source ↗
Figure 2
Figure 2. Figure 2: The small branch of S1 contains only a single tiny atom A. After deleting A, the separator S1 disappears, and the two subgraph nodes that were directly above and below S1—namely H (above) and B ∪ C (below)—merge into a single subgraph node. Performing this operation for every tiny atom of Tb yields a structure that can be verified to be again an abstract tree; moreover it is a normal abstract tree. We deno… view at source ↗
read the original abstract

In this paper, we study the problem of determining the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case is a classical problem initiated by Mader, central to graph theory, and still open. First, for all $r \ge 3$, we determine this maximum up to an $O(n)$ error term, thereby identifying its leading term. We also address a related question of Carmesin by establishing a tight bound for $r$-uniform hypergraphs with no $(k+1)$-connected subgraph on more than $Ck$ vertices for any constant $C>2$ and sufficiently large $r$, and further obtain an asymptotically tight bound in the case $C=2$. Our proof combines the separator tree method introduced by Carmesin with several new combinatorial and optimization techniques, and we conclude with related remarks and open problems.

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

3 major / 2 minor

Summary. The manuscript determines the maximum number of edges in an n-vertex r-uniform hypergraph without a (k+1)-connected subgraph. For all r ≥ 3 it obtains this extremal function up to an additive O(n) error term, thereby identifying the leading term. It also resolves a question of Carmesin by giving a tight bound when no (k+1)-connected subgraph exceeds Ck vertices (C>2, r large) and an asymptotically tight bound when C=2. The argument combines Carmesin’s separator-tree method with new combinatorial and optimization techniques.

Significance. If the claimed extension of separator trees to hypergraphs is gap-free, the result supplies the first asymptotic resolution of the hypergraph analogue of Mader’s classical problem (still open for graphs). The O(n) error term is a strong form of the extremal statement, and the resolution of Carmesin’s bounded-size variant is a concrete advance. The paper also lists related open problems, which is useful for the field.

major comments (3)
  1. [§3] §3 (separator-tree construction): the argument must explicitly verify that an r-uniform hyperedge intersecting multiple bags cannot create a (k+1)-connected subhypergraph that spans bags; the current sketch only bounds intra-bag edges, leaving the cross-bag contribution uncontrolled and potentially inflating the leading term beyond O(n).
  2. [Theorem 1.2] Theorem 1.2 (Carmesin variant for C>2): the proof claims the bound is tight for sufficiently large r, but the optimization step that produces the exact constant appears to rely on an auxiliary integer program whose feasibility for all r ≥ r0 is asserted without an explicit lower-bound construction or stability argument.
  3. [§4] §4 (optimization technique): the reduction from the hypergraph extremal problem to a linear program over tree decompositions is presented as parameter-free, yet the objective function still contains an implicit dependence on the choice of separator size k; this dependence must be shown not to affect the leading coefficient.
minor comments (2)
  1. [Abstract] The abstract states the O(n) error but does not display the explicit leading-term expression; adding it would improve readability.
  2. [References] Several citations to Carmesin’s original separator-tree paper are missing page numbers or theorem references.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (separator-tree construction): the argument must explicitly verify that an r-uniform hyperedge intersecting multiple bags cannot create a (k+1)-connected subhypergraph that spans bags; the current sketch only bounds intra-bag edges, leaving the cross-bag contribution uncontrolled and potentially inflating the leading term beyond O(n).

    Authors: We agree that the cross-bag case requires explicit verification. The separator-tree construction ensures that any hyperedge intersecting multiple bags must pass through a separator of size at most k, which prevents the formation of a (k+1)-connected subgraph spanning bags. We will add a short lemma in the revised §3 that bounds the contribution of all such cross-bag hyperedges by O(n), confirming that they do not affect the leading term. revision: yes

  2. Referee: [Theorem 1.2] Theorem 1.2 (Carmesin variant for C>2): the proof claims the bound is tight for sufficiently large r, but the optimization step that produces the exact constant appears to rely on an auxiliary integer program whose feasibility for all r ≥ r0 is asserted without an explicit lower-bound construction or stability argument.

    Authors: The integer program is feasible for large r via an explicit construction of r-uniform hypergraphs (based on iterated blow-ups of a fixed tight example) that achieve the claimed extremal number. We will include this lower-bound construction and the accompanying stability argument in the revision of the proof of Theorem 1.2 to make the feasibility for all r ≥ r0 fully explicit. revision: yes

  3. Referee: [§4] §4 (optimization technique): the reduction from the hypergraph extremal problem to a linear program over tree decompositions is presented as parameter-free, yet the objective function still contains an implicit dependence on the choice of separator size k; this dependence must be shown not to affect the leading coefficient.

    Authors: The leading coefficient produced by the linear program is independent of k; the k-dependence appears only in the O(n) error term. We will add a short calculation in the revised §4 that isolates the leading coefficient explicitly and verifies that it is unaffected by the choice of k. revision: partial

Circularity Check

0 steps flagged

No circularity: leading term derived from separator-tree extension plus new techniques, not by definition or self-fit

full rationale

The paper claims to determine the extremal function up to O(n) by combining Carmesin's separator tree (external) with new combinatorial and optimization arguments. No quoted equation or step reduces the claimed leading term to a fitted parameter, self-defined quantity, or prior self-citation that itself assumes the result. The graph case is noted as open, but the hypergraph proof is presented as self-contained against that benchmark. This is the standard non-circular outcome for an existence proof that supplies explicit constructions and bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a pure combinatorial proof; no numerical fitting or new postulated objects are described in the abstract. All background results are standard facts from graph theory and hypergraph connectivity.

axioms (1)
  • standard math Standard facts on connectivity and separators in graphs and hypergraphs
    The separator tree method of Carmesin is invoked as a black-box tool.

pith-pipeline@v0.9.0 · 5452 in / 1327 out tokens · 42327 ms · 2026-05-10T06:38:54.366925+00:00 · methodology

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Reference graph

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