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arxiv: 2605.22397 · v1 · pith:7UGXBNC3new · submitted 2026-05-21 · 🧮 math.CO

On the Tur\'an number of blow-ups of mathcal{F}₅

Xiamiao Zhao , Xin Cheng , D\'aniel Gerbner , Hilal Hama Karim , Shujing Miao , Yichen Wang , Junpeng Zhou This is my paper

Pith reviewed 2026-05-22 04:25 UTC · model grok-4.3

classification 🧮 math.CO
keywords Turán number3-uniform hypergraphsblow-upsF5 hypergraphextremal constructionsstabilitysupersaturation
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The pith

The Turán number of the f3-blow-up of F5 is exactly determined and admits exponentially many extremal constructions.

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

The paper determines the exact value of the Turán number ex(n, F5(f3;t)) for the 3-uniform hypergraph obtained by replacing vertex f3 in F5 with t new vertices. It shows that this number is achieved by exponentially many distinct hypergraphs for infinitely many t, rather than only the balanced complete 3-partite construction. General upper and lower bounds are given for the Turán number of any blow-up of F5, with exact values recovered for special cases such as t disjoint copies of F5. A new auxiliary hypergraph F_sim(t) is constructed that is contained in any hypergraph obtained by adding an edge to the Turán hypergraph yet has a different family of extremal examples.

Core claim

For the hypergraph F5(f3;t) formed by blowing up vertex f3 of F5 to t vertices, the Turán number ex(n, F5(f3;t)) equals the number of edges in a certain balanced 3-partite construction plus lower-order terms that depend on t, and this extremal value is attained by exponentially many non-isomorphic hypergraphs when t belongs to an infinite arithmetic progression.

What carries the argument

The blow-up F5(f3;t) obtained by replacing f3 with an independent set of size t, together with the stability and supersaturation arguments transferred from the earlier f5-blow-up case.

If this is right

  • For infinitely many t the extremal function ex(n, F5(f3;t)) is achieved by 2^{\Theta(n^2)} distinct constructions.
  • The same general upper and lower bounds apply to every vertex blow-up of F5.
  • Exact Turán numbers are obtained for the t-fold disjoint union of copies of F5.
  • The auxiliary hypergraph F_sim(t) has a Turán number strictly larger than that of the balanced complete 3-partite hypergraph.

Where Pith is reading between the lines

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

  • Changing which vertex is blown up can preserve the exponential multiplicity of extremal examples while altering the precise density.
  • The same phenomenon may appear for blow-ups of other 3-uniform hypergraphs that already possess multiple extremal constructions.
  • It would be natural to check whether the exponential number of constructions persists when t grows with n.

Load-bearing premise

Stability and supersaturation methods that worked for the f5-blow-up continue to apply without new structural obstructions when the blown-up vertex is f3 instead.

What would settle it

An n-vertex 3-uniform hypergraph with more than the claimed number of edges that still avoids F5(f3;t) for some large n and t in the arithmetic progression, or a count showing only polynomially many extremal hypergraphs rather than exponentially many.

Figures

Figures reproduced from arXiv: 2605.22397 by D\'aniel Gerbner, Hilal Hama Karim, Junpeng Zhou, Shujing Miao, Xiamiao Zhao, Xin Cheng, Yichen Wang.

Figure 3
Figure 3. Figure 3: The unique extremal graph of F T3(m, 3)+E (i.e. adding another hyperedge to T3(m, 3)) for some integer m. This condition is clearly necessary, and Simonovits [21] showed that in the graph case, the analogous condition is also sufficient. However, Balogh, Clemen, and Luo [1] gave a complicated counterexample, showing that this condition is not sufficient. Here, we present a simpler counterexample. Let Fsim(… view at source ↗
Figure 4
Figure 4. Figure 4: The vertices {ai} m i=1 ∪ {bi} m i=1 ∪ {ci} m i=1 and D B,1 1 form a copy of F Let us delete the at most N1 vertices described in Claim 3.8 from A′ L and let R1 denote the resulting set, i.e. the set of vertices v ∈ A′ L with |G3(v)|< 1 10 δn2 . According to (3) and the bounds on |G2(v)|, for every v ∈ R1, we have |GB 4 (v)|+|GC 4 (v)|≥ δn2 − |G2(v)|−|G3(v)|≥ δn2 − 3 2 δ 2n 2 − 1 10 δn2 ≥ δ 2 n 2 , when n … view at source ↗
Figure 5
Figure 5. Figure 5: There exist s vertices in D B,2 2 and s vertices in W1, together with T ∪ WB ∪ WC, they form a copy of F. which implies that |G1(v)|≥ 1 3 (|B|·|C|−3δn2 ). According to the definition of A′ , for each u ∈ W1, there are at most 2δn2 pairs (b, c) ∈ B × C such that ubc ̸∈ E(G). It implies that at least |B||C|−2mδn2 pairs (b, c) ∈ B × C satisfy that ubc ∈ E(G) for all u ∈ W1. We construct an auxiliary bipartite… view at source ↗
Figure 6
Figure 6. Figure 6: The case when there exists a hyperedge containing three vertices from [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: ) Symmetrically, we can prove that {a1, . . . , at , v} ∩ A′ ̸= ∅. ■ Claim 5.2. We have |E2|≤ 6δ · |M|+|S t (|A|)|+S t (|B|) + S t (|C|). Proof. We only give an upper bound on the size of E A 2 , since the other two cases are similar. Let E A′ ≥2 2 = {E ∈ EA 2 : |E ∩ A′ |≥ 2}. For every u ∈ A′ , note that dM(u) ≥ 2δn2 , and the number of hyperedges in E A′ ≥2 2 that contain u is at most |A′ |·n ≤ 1 2 δ 2n … view at source ↗
Figure 7
Figure 7. Figure 7: The vertices {u, v, b, c} and {ai} t i=1 forms a copy of F5(f3;t) is no vertex in Lu with degree at least t. Otherwise, there exists a t-sunflower containing u such that the other t + 1 vertices are in A \ A′ , a contradiction with Claim 5.1. As a result, e(Lu) ≤ (t − 1)n. Then we have |EA′ =1 2 | |M| ≤ (t − 1)n 2δn2 < δ. (11) According to Claim 5.1, there is no t-sunflower with hyperedges in E A 2 \ ((E A… view at source ↗
read the original abstract

Let $\mathcal{F}_5$ denote the $3$-uniform hypergraph on the vertex set $\{f_1,f_2,\dots,f_5\}$ with hyperedges $\{f_1f_2f_3,f_1f_2f_4,f_3f_4f_5\}$. Recently, Balogh, Clemen and Luo determined the Tur\'an number of a one-vertex blow-up of $\mathcal{F}_5$, more specifically, they blow up the vertex $f_5$ to $t$ vertices, the resulting hypergraph is denoted by $\mathcal{F}_5(f_5;t)$. They show that for infinitely many $t$, $\mathcal{F}_5(f_5;t)$ has exponentially many extremal constructions and positive Tur\'an density. In this paper, we determine the exact Tur\'an number of the hypergraph obtained by blowing up $f_3$ of $\mathcal{F}_5$ to $t$ vertices and show that it also has exponentially many extremal constructions. We also give a general upper bound and lower bound of the Tur\'an number of every blow-up of $\mathcal{F}_5$. For some special blow-ups of $\mathcal{F}_5$, for example, $t$-disjoint copies of $\mathcal{F}_5$, we determine the exact Tur\'an number. We construct a hypergraph $\mathcal{F}_{sim}(t)$ which is a subgraph of a blow-up of $\mathcal{F}_5$, and is contained in the hypergraph obtained by adding any new hyperedge to the Tur\'an hypergraph (the balanced complete $3$-partite hypergraph), but its extremal construction is not the Tur\'an hypergraph. We also determine the exact Tur\'an number of $\mathcal{F}_{sim}(t)$.

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

1 major / 2 minor

Summary. The manuscript determines the exact Turán number ex(n, F5(f3;t)) for the 3-uniform hypergraph obtained by blowing up vertex f3 of F5 to t vertices, proves the existence of exponentially many extremal constructions for infinitely many t, gives general upper and lower bounds on the Turán number for arbitrary blow-ups of F5, determines exact values for special cases such as t-disjoint copies of F5, and introduces the auxiliary hypergraph F_sim(t) (a subgraph of a blow-up contained in any hypergraph obtained by adding an edge to the balanced complete 3-partite hypergraph) for which the exact Turán number is also determined.

Significance. If the proofs are correct, the work extends the Balogh-Clemen-Luo results on the f5-blow-up by establishing analogous exact results and multiple extremal examples for the f3-blow-up position, while the general bounds and the F_sim(t) construction supply new examples of non-Turán extremal hypergraphs in 3-uniform settings. The explicit treatment of special blow-ups and the non-Turán example strengthen the contribution to hypergraph Turán problems with positive density.

major comments (1)
  1. [stability lemma and main theorem for F5(f3;t)] The stability and supersaturation arguments for the f3-blow-up case rely on direct adaptation of the methods developed for the f5-blow-up. However, f3 participates in two edges of the base F5 while f5 participates in only one; this alters the link graphs and admissible overlapping configurations after blow-up. The manuscript must verify explicitly that no additional extremal families survive the case analysis, as this verification is load-bearing for both the exact value and the exponential count of constructions.
minor comments (2)
  1. [Abstract] The abstract introduces the notation F5(f3;t) without a one-sentence reminder of the edge set of the base F5; adding this would improve standalone readability.
  2. [Introduction] Notation for the general blow-up F5(v;t) could be fixed earlier in the introduction to streamline the transition from the f5 case to the f3 case and to the general bounds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [stability lemma and main theorem for F5(f3;t)] The stability and supersaturation arguments for the f3-blow-up case rely on direct adaptation of the methods developed for the f5-blow-up. However, f3 participates in two edges of the base F5 while f5 participates in only one; this alters the link graphs and admissible overlapping configurations after blow-up. The manuscript must verify explicitly that no additional extremal families survive the case analysis, as this verification is load-bearing for both the exact value and the exponential count of constructions.

    Authors: We thank the referee for highlighting the structural distinction between the f3 and f5 positions. In the manuscript, the stability lemma and case analysis for F5(f3;t) have been modified to reflect f3's participation in two base edges. This leads to adjusted link graphs and a broader set of admissible overlaps, which are enumerated explicitly in Section 3. The supersaturation argument is strengthened by this double incidence, as deviations create additional copies of F5(f3;t) more readily than in the f5 case. The case analysis rules out all other configurations, ensuring no additional extremal families exist. To address the request for more explicit verification, we will add a clarifying paragraph in the revised version summarizing why the enumerated overlaps exhaust all possibilities. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external combinatorial stability results

full rationale

The paper determines exact Turán numbers ex(n, F5(f3;t)) for the f3-blow-up using stability and supersaturation arguments transferred from the prior Balogh-Clemen-Luo result on the distinct f5-blow-up. No equation or construction in the provided text reduces by definition to a fitted parameter or to a quantity defined in terms of the target result itself. The referenced prior work is by different authors and concerns a non-identical blow-up (f5 lies in one edge while f3 lies in two), so the transfer is an external assumption rather than a self-citation chain or self-definitional loop. General bounds and special cases (t-disjoint copies, F_sim(t)) are likewise established by direct counting and construction arguments without renaming known patterns or smuggling ansatzes via self-citation. This is the normal case of an independent combinatorial proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely within standard combinatorial frameworks and does not introduce new free parameters, invented entities, or non-standard axioms.

axioms (1)
  • standard math Standard stability and supersaturation results from extremal hypergraph theory for 3-uniform hypergraphs.
    Likely invoked to transfer known behavior of the base F5 to the blown-up versions.

pith-pipeline@v0.9.0 · 5895 in / 1338 out tokens · 51666 ms · 2026-05-22T04:25:32.491990+00:00 · methodology

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Reference graph

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