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On tight tree-complete hypergraph Ramsey numbers

T0 review · 1 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For r=3 or 4, every non-trivial tight r-tree has hypergraph Ramsey number $\Theta(n^{r-1})$.

desk verdict New Theta(n^{r-1}) bounds for tight-tree hypergraph Ramsey numbers for r=3,4, with a repairable proof gap in the lower-bound lemmas. read the letter →

arxiv 2412.19461 v1 pith:GHIJ2HUZ submitted 2024-12-27 math.CO

Jiaxi Nie This is my paper · ORCID
classification math.CO MSC 05C5505C65
keywords RamseynumberstighttreeshypergraphsindependentsetsCooper-MubayiconstructiongridChvataltheorem
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

The paper proves a hypergraph analogue of the classical graph-tree Ramsey theorem for uniformities 3 and 4. If $T$ is a non-trivial tight $r$-tree, then the Ramsey number $R(T,n)$ is bounded between two positive constants times $n^{r-1}$, for $r=3$ or $4$. The upper bound follows from a random deletion argument applied to a folklore extremal estimate on $T$-free hypergraphs. The lower bound is the main work: for $r=3$ it uses a known L-shape construction on an $n\times n$ grid, and for $r=4$ it introduces a new construction on the $n^3$ grid whose edges are quadruples radiating from a center along three coordinate-directed orders. The author conjectures the same order of magnitude for every $r\ge 5$, while noting that the natural signed-lexicographic generalization fails at $r=5$.

What carries the argument

The load-bearing object is the 'center' of an edge. In both constructions, every edge is a set of $r$ grid points $\{P_0,P_1,\dots,P_{r-1}\}$ with a distinguished center $P_0$, and the defining inequalities make the center the unique point from which the other $r-1$ points are reached along prescribed directed coordinate orders. Propositions 3.2 and 4.3 show that in $H_3$ and $H_4$ any two edges intersecting in $r-1$ vertices have the same center; this is the property that forbids non-trivial tight trees. For the 4-uniform case, the three directed graphs $T_1,T_2,T_3$ on $[n]^3$ are defined by signed lexicographic inequalities in coordinate pairs, and the proof of the center property is a four-case analysis using lexicographic height functions such as $F_{21}(P)=P(2)(n+1)+P(1)$ and $F_{2(-1)(-3)}(P)=P(2)(n+1)^2-P(1)(n+1)-P(3)$.

What would settle it

A small exhaustive search, e.g., over $[4]^3$ or $[5]^3$, for two distinct edges of $H_4$ that share exactly three points but have different centers would directly disprove Proposition 4.3. If such a pair exists, the construction contains a non-trivial tight 4-tree and the claimed $\Theta(n^3)$ lower bound for non-trivial tight 4-trees is false.

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Extended reading notes

Core claim

Theorem 1.1 is the central discovery: for $r=3$ or $4$, if $T$ is a non-trivial tight $r$-tree, then there exist constants $c_1,c_2>0$ such that $c_1 n^{r-1}\le R(T,n)\le c_2 n^{r-1}$. The proof isolates the obstruction to a tight tree inside the host hypergraph: in the 3-uniform L-shape graph on $[n]^2$ and in the new 4-uniform graph on $[n]^3$, any two edges that meet in $r-1$ vertices must share the same distinguished center vertex. Since consecutive edges in any tight-tree ordering meet in $r-1$ vertices, every embedded tight tree is forced to be trivial, meaning all its edges share a common vertex. The new 4-uniform construction therefore has independence number at most $10n$ and no non-trivial tight 4-tree, which supplies the missing cubic lower bound.

Load-bearing premise

The 4-uniform lower bound rests on Proposition 4.3's four-case coordinate-inequality check that any two edges sharing three vertices have the same center; if that check omits a configuration, the construction could host a non-trivial tight 4-tree and the cubic lower bound would fail.

Editorial extensions

If this is right

  • For every non-trivial tight 3- or 4-tree, $R(T,n)$ is within a constant factor of $n^{r-1}$, so the Ramsey number is polynomial of degree exactly $r-1$ in $n$.
  • The new 4-uniform construction provides the first example of a dense 4-graph with no non-trivial tight 4-tree and small independence number, directly giving the lower bound in Theorem 1.1.
  • Combining the construction with the random-deletion upper bound yields explicit constants: for a tight $r$-tree with $k$ edges, $R(T,n)\le 2(k-1)(2en/(r-1))^{r-1}$.
  • The result separates non-trivial tight trees from two-edge tight trees, for which the Ramsey number is known to be $\Theta(n^{r-1}/\log n)$.

Reading between the lines

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

  • The constant $10n$ in Proposition 4.2 is probably not optimal; a sharper deletion argument or a different auxiliary set system could lower it, which would improve the constant in the $n^3$ lower bound.
  • The center property suggests a general recipe for tight-tree lower bounds in uniformity $r$: build an $r$-graph on a product grid so that edges sharing $r-1$ vertices have a common center, and prove a density lemma forcing an edge in every large set.
  • The paper's computer check on 5-uniform signed-lexicographic variants indicates that the lexicographic template exhausts itself at $r=4$; a proof for $r\ge 5$ will likely need a genuinely different intersection structure rather than more coordinates.
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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 / 6 minor

Summary. The paper studies hypergraph Ramsey numbers R(T,n) for r-uniform tight trees T. For r=3 and r=4, it is shown that for every non-trivial tight r-tree T, R(T,n)=Theta(n^{r-1}). The upper bound is derived by random deletion from a known edge-count bound for tight trees. The lower bound for r=3 uses a construction of Cooper and Mubayi on [n]^2, and the main novelty is a 4-uniform construction on [n]^3 whose edges are determined by three tournaments T1,T2,T3. The paper also gives a computer-verified note that a natural 5-uniform generalization fails.

Significance. If correct, the result is a natural hypergraph analogue of Chvatal's theorem, establishing the expected order of magnitude in the first two non-trivial uniformities and suggesting a general conjecture. The constructions are explicit and elementary, requiring no heavy machinery. The paper is clearly written and carefully checks a plausible obstruction for r=5. The 4-uniform case analysis in Proposition 4.3 is intricate and is the main technical burden; I did not find an overlooked configuration in that case analysis. The lower bounds are proved by explicit constructions, and the upper bound uses a cited edge-count bound, so there is no circularity or parameter fitting.

major comments (1)
  1. [§3 (Proposition 3.3) and §4 (Proposition 4.4)] The proofs of Propositions 3.3 and 4.4 contain a false assertion. They state that for an edge ordering satisfying Definition (1), every pair of consecutive edges e_i and e_{i+1} intersect in r-1 vertices. This is not implied by (1), which only requires each new edge to attach to some earlier edge, not to the immediately preceding one. For example, in a tight 4-tree with e1={1,2,3,4}, e2={1,2,3,5}, e3={1,2,4,6}, the order e1,e2,e3 satisfies (1) but |e2∩e3|=2<3. Since Propositions 3.3 and 4.4 are used to prove the lower bounds, this is a load-bearing gap. The gap is repairable: for each i≥2, if e_s is the parent edge from (1) and v is the new vertex, then |e_i∩e_s|=r-1; induction over the parent relation (which is well-founded because s<i) gives center(e_i)=center(e_s) and hence all edges share the center of e_1. I recommend replacing the consecutive-edge argument with this induction.
minor comments (6)
  1. [§4 (Proposition 4.2)] The deletion process should be stated more explicitly: it must be repeated until no X_ij has size at most 3, and the count of deleted vertices should be justified by noting that each of the 3n sets is deleted at most once, so the total number of deleted vertices is at most 9n. As written, the assertion that every nonempty final slice has size at least 4 does not follow from a single pass.
  2. [§4 (Proposition 4.3, Case 2)] In the paragraph before 'Moreover, recall Equation (2)', the expression 'F21(P')' should be 'F21(P'_0)'; the current text uses an undefined symbol P'.
  3. [§4 (Proposition 4.3)] The phrase 'Up to symmetry' is not self-explanatory. The four cases cover the situation after possibly swapping the two edges e and e' (so that the center of e' lies in e or neither center lies in the other edge), and the text should say this explicitly.
  4. [§3 (Proposition 3.1)] The proof asserts that in the cycle 'there must be three consecutive edges c_{j'} r_i, r_i c_j, c_j r_{i'} such that i' > i and j' > j'. Please add a one-sentence justification (e.g., orient the cycle so that a step with row increase and column decrease occurs; such a step must exist in any simple cycle).
  5. [§2 (Proposition 2.2)] There is a typo: 'comletes' should be 'completes'.
  6. [§5 (Concluding remarks)] In the first bullet point, '2 33 ! = 48' should read '2^3 * 3! = 48' (the current typesetting appears to show 2^(33!), which is incorrect).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower bounds are explicit hypergraph constructions and the upper bound uses an independent cited edge-count bound.

full rationale

The paper's derivation chain is self-contained with respect to the target theorem. The upper bound in Theorem 1.1 follows from Proposition 2.2, which uses the random deletion method together with Proposition 2.1, a cited edge-count bound for tight trees from Furedi and Jiang [6]; it does not assume the Ramsey bound being proved. The 3-uniform lower bound rests on the explicit Cooper-Mubayi construction H3 on [n]^2 and on Propositions 3.1-3.3, which show H3 is H3-free of non-trivial tight 3-trees. The 4-uniform lower bound rests on the new explicit construction H4 on [n]^3 and on Propositions 4.2-4.4, including a direct four-case coordinate-inequality proof of Proposition 4.3. There are no fitted parameters, no equation defined in terms of the target result, and no self-citation chain that supplies the central content; the cited results are external and their assumptions do not include the theorem. The skeptical note about the consecutive-edge assertion in Propositions 3.3 and 4.4 concerns a proof-technique gap that is repairable by induction over the parent-edge relation; it is a correctness risk, not a circularity, because no step reduces the theorem to its own statement or to a fitted input. The concluding computer-checked remark about 5-uniform generalizations is not used as evidence for the r=3,4 claims. Overall circularity score: 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper's central claim relies on a cited edge-count bound for T-free hypergraphs, standard graph-theoretic facts, and the definition of tight trees. No free parameters are fitted; the constructions use only the grid size n. No new physical or mathematical entities are postulated beyond the explicit hypergraph constructions.

assumptions (3)
  • domain assumption An n-vertex r-graph with no tight r-tree T having k edges has at most (k-1)*C(n,r-1) edges.
    Cited as Proposition 5.4 of Furedi and Jiang [6] and used in Proposition 2.2 to derive the upper bound; not proved in the paper.
  • standard math Any bipartite graph with at least as many edges as vertices contains a cycle.
    Used in Proposition 3.1 to find an L-shape in any 2n-element subset of the grid.
  • standard math In any tournament, at most one vertex has outdegree zero.
    Used in Proposition 4.2 to bound the size of the set Bi of vertices with zero outdegree in each Ti.

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Pith. "Pith review of On tight tree-complete hypergraph Ramsey numbers." pith.science (2026). https://pith.science/paper/GHIJ2HUZ

@misc{pith2026241219461,
  author       = {Pith},
  title        = {Pith review of: On tight tree-complete hypergraph Ramsey numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GHIJ2HUZ}},
  note         = {Machine review of arXiv:2412.19461}
}
abstract

Chv\'atal showed that for any tree $T$ with $k$ edges the Ramsey number $R(T,n)=k(n-1)+1$ ("Tree-complete graph Ramsey numbers." Journal of Graph Theory 1.1 (1977): 93-93). For $r=3$ or $4$, we show that, if $T$ is an $r$-uniform non-trivial tight tree, then the hypergraph Ramsey number $R(T,n)=\Theta(n^{r-1})$. The 3-uniform result comes from observing a construction of Cooper and Mubayi. The main contribution of this paper is the 4-uniform construction, which is inspired by the Cooper-Mubayi 3-uniform construction.

Figures

Figures reproduced from arXiv: 2412.19461 by the authors.

Figure 1
Figure 1. P0,P1 and P2 form an edge in H3 Proposition 3.1. Every 2n-element set in [n] 2 contains an edge in H3. Proof. Let X be a 2n-element set in [n] 2 . Consider a bipartite graph B on R ⊔ C where R = {r1, r2, . . . , rn} and C = {c1, c2, . . . , cn} such that ri and cj form an edge if and only if (i, j) ∈ X. We can view R and C as the set of rows and the set of columns of the n × n grid respectively. Note that the number… view at source ↗
Figure 2
Figure 2. is a visual illustration of T1, which depicts the possible positions of Q relative to P. P 2 nd coordinate 3 rd coordinate [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. is a visual illustration of T2, which depicts the possible positions of Q relative to P. P 1 st coordinate 3 rd coordinate [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Here the s are the possible positions of Q if P Q ∈ T3 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

  1. [1]

    Alon and J

    N. Alon and J. H. Spencer. The probabilistic method . John Wiley & Sons, 2016

    work page 2016

  2. [2]

    Chv´ atal

    V. Chv´ atal. Tree-complete graph ramsey numbers. Journal of Graph Theory , 1(1):93–93, 1977

    work page 1977

  3. [3]

    Cooper and D

    J. Cooper and D. Mubayi. Sparse hypergraphs with low independe nce number. Combinatorica, 37:31–40, 2017

    work page 2017

  4. [4]

    P. Erd¨ os. Extremal problems in graph theory. Publ. House Cszechoslovak Acad. Sci., Prague , pages 29–36, 1964

    work page 1964

  5. [5]

    Frankl and Z

    P. Frankl and Z. F¨ uredi. Exact solution of some tur´ an-type problems. Journal of Combinatorial Theory, Series A , 45(2):226–262, 1987

    work page 1987

  6. [6]

    F¨ uredi and T

    Z. F¨ uredi and T. Jiang. Tur \’an numbers of hypergraph trees. arXiv preprint arXiv:1505.03210 , 2015

    arXiv 2015

  7. [7]

    F¨ uredi, T

    Z. F¨ uredi, T. Jiang, A. Kostochka, D. Mubayi, and J. Verstra¨ ete. Hypergraphs not containing a tight tree with a bounded trunk. SIAM Journal on Discrete Mathematics , 33(2):862–873, 2019

    work page 2019

  8. [8]

    F¨ uredi, T

    Z. F¨ uredi, T. Jiang, A. Kostochka, D. Mubayi, and J. Verstra¨ ete. Hypergraphs not containing a tight tree with a bounded trunk ii: 3-trees with a trunk of size 2. Discrete Applied Mathematics , 276:50–59, 2020

    work page 2020

Show all 11 references
  1. [9]

    F¨ uredi, T

    Z. F¨ uredi, T. Jiang, A. Kostochka, D. Mubayi, and J. Verstra¨ ete. Tight paths in convex geometric hypergraphs. Advances in Combinatorics , 1, 2020

    2020

  2. [10]

    K. T. Phelps and V. R¨ odl. Steiner triple systems with minimum indep endence number. Ars Combin , 21:167–172, 1986

    1986

  3. [11]

    R¨ odl and E.ˇSinajov´ a

    V. R¨ odl and E.ˇSinajov´ a. Note on independent sets in steiner systems.Random Structures & Algorithms , 5(1):183–190, 1994. 8

    1994

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