A note on hypergraphs with asymmetric Ramsey properties
Pith reviewed 2026-05-21 03:56 UTC · model grok-4.3
The pith
There exists an r-graph G that avoids forcing monochromatic K_ti in all colors but forces a version with s = R minus one and the last color reduced by one.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The note proves that for any integers t1 ≥ ⋯ ≥ tℓ > r there exists an r-graph G such that G does not arrow to (K^{(r)}_{t1}, …, K^{(r)}_{tℓ}) but G does arrow to (K^{(r)}_s, K^{(r)}_{tℓ-1}), where s equals the Ramsey number R(K^{(r)}_{t1}, …, K^{(r)}_{tℓ}) minus one.
What carries the argument
The classical multi-color Ramsey number R(K^{(r)}_{t1}, …, K^{(r)}_{tℓ}) that supplies the integer s and anchors the construction of the auxiliary r-graph G with the required asymmetric arrowing properties.
If this is right
- The complete multi-color Ramsey property can be avoided while a nearby weakened version remains satisfied.
- Such separating r-graphs exist for every parameter tuple obeying the stated inequalities.
- The same separation holds at every uniformity level r ≥ 2.
- These examples distinguish distinct strengths of Ramsey forcing in ℓ-edge-colorings of hypergraphs.
Where Pith is reading between the lines
- The same auxiliary-graph technique may apply to Ramsey questions for non-complete forbidden subhypergraphs.
- Minimal-size versions of the constructed G could yield new upper bounds on certain asymmetric Ramsey numbers.
- Analogous separations might appear in canonical Ramsey theorems or in hypergraph Turán problems.
- The result suggests that the landscape of Ramsey arrows for hypergraphs is finer than the single threshold given by R alone.
Load-bearing premise
The classical multi-color Ramsey number R for the given complete r-graphs is finite and an auxiliary r-graph can be built from it without circular dependence on the claimed existence statement.
What would settle it
A concrete counterexample would be any specific r, ℓ and tuple t1 ≥ ⋯ ≥ tℓ > r for which every r-graph that arrows to (K_s, K_{tℓ-1}) also arrows to the full tuple (K_{t1}, …, K_{tℓ}).
read the original abstract
Let $r,\ell\geq2$ be integers. Given $r$-graphs $G$ and $F_1,\dots,F_\ell$, we write $G\to(F_1,\dots,F_\ell)$ if every $\ell$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$th color for some $1\leq i\leq\ell$, otherwise we write $G\not\to(F_1,\dots,F_\ell)$. The Ramsey number $R(F_1,\dots,F_\ell)$ is the minimum number of vertices in an $r$-graph $G$ satisfying $G\to(F_1,\dots,F_\ell)$. In this note we prove that for any integers $t_1\geq\dots\geq t_\ell>r$, there exists an $r$-graph $G$ such that $G\not\to(K^{(r)}_{t_1},\dots,K^{(r)}_{t_\ell})$ but $G\to(K^{(r)}_s,K^{(r)}_{t_\ell-1})$, where $s=R(K^{(r)}_{t_1},\dots,K^{(r)}_{t_\ell})-1$. This extends recent work by Mendon\c{c}a, Miralaei, and Mota, who established the statement for $r=2$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for integers r, ℓ ≥ 2 and t1 ≥ ⋯ ≥ tℓ > r, letting R = R(K_{t1}^{(r)}, …, K_{tℓ}^{(r)}), there exists an r-graph G on s = R − 1 vertices such that G does not arrow the ℓ-tuple (K_{t1}^{(r)}, …, K_{tℓ}^{(r)}) but does arrow the 2-tuple (K_s^{(r)}, K_{tℓ−1}^{(r)}). This extends the r = 2 case due to Mendonça, Miralaei, and Mota. The argument relies on the classical finiteness of the hypergraph Ramsey number R together with an auxiliary existence construction on s vertices.
Significance. If the result holds, it demonstrates that hypergraph Ramsey numbers admit asymmetric witnesses on the critical order s = R − 1: the same vertex set can avoid the full multicolour Ramsey property while still forcing a different asymmetric pair. This clarifies the boundary between symmetric and asymmetric Ramsey behaviour for r-uniform hypergraphs and supplies concrete examples beyond the graph case. The proof inherits finiteness from the classical theorem and adds no new circularity.
minor comments (2)
- §2, Definition 1.1: the arrowing notation G → (F1, …, Fℓ) is introduced without an explicit reminder that the colours are ordered; a parenthetical note would prevent misreading when the target tuple is asymmetric.
- The auxiliary construction establishing the existence of G on s vertices that arrows (K_s^{(r)}, K_{tℓ−1}^{(r)}) is only sketched; expanding the inductive or probabilistic step into a short paragraph would improve readability for readers unfamiliar with the r = 2 precedent.
Simulated Author's Rebuttal
We thank the referee for their positive assessment and recommendation to accept the paper. The referee's summary correctly identifies the main contribution: an existence result for r-graphs on s = R-1 vertices that witness an asymmetric Ramsey property while avoiding the full multicolour arrowing.
Circularity Check
No significant circularity; existence claim is independent of its inputs
full rationale
The paper defines R(F1,...,Fℓ) explicitly as the smallest order of any r-graph G satisfying the arrowing property G→(F1,...,Fℓ). The stated theorem then asserts existence of some G on exactly s = R−1 vertices that fails the full ℓ-tuple arrowing (true for every graph on s vertices by minimality of R) while satisfying the 2-tuple arrowing G→(Ks, K_{tℓ−1}). The non-trivial content is therefore the auxiliary existence argument for the 2-tuple property, which the paper supplies by extending the r=2 construction of Mendonça et al. Finiteness of R is taken from the classical hypergraph Ramsey theorem and introduces no self-reference. No equation, parameter fit, or self-citation reduces the new existence statement to a tautology or to the input data; the derivation remains self-contained.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Ramsey numbers R(K^{(r)}_{t1},…,K^{(r)}_{tℓ}) exist and are finite for the given parameters
Reference graph
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discussion (0)
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