Unordered Canonical Ramsey Numbers

• Unordered canonical Ramsey numbers are defined as the smallest n for which every edge-coloring of Kn contains either an orderable clique of size s or a rainbow clique of size t.
• Recent studies have precisely determined small values using computational methods including ILP, exhaustive enumeration, and tabu search.
• Advanced techniques like flag algebras and probabilistic analysis have refined asymptotic bounds, revealing polynomial growth rates for fixed s as t increases.

Unordered canonical Ramsey numbers, denoted $CR(s, t)$, generalize classical Ramsey theory to colored complete graphs lacking any pre-imposed vertex ordering, and seek the threshold $n$ so that every edge-coloring of $K_n$ contains either an “orderable” clique of size $s$ or a rainbow clique of size $t$. This concept, introduced by Richer, refines the canonical Ramsey numbers of Erdős–Rado by emphasizing edge-colorings whose structure can be interpreted through suitable vertex labelings. Recent advances have precisely determined several small cases, established sharp polynomial and asymptotic bounds for larger parameters, developed computational approaches for extremal constructions, and exposed rich connections to probabilistic, algebraic, and combinatorial methodology.

1. Formal Definition and Canonical Structures

For positive integers $s, t \ge 3$, let $K_n$ denote the complete graph on $n$ vertices and $\chi: E(K_n) \to \Omega$ a coloring of its edges using an arbitrary palette $\Omega$. A subgraph $K_s$ is called orderable under $\chi$ if there exists a labelling $v_1,\dots,v_s$ of its vertices such that for any two edges $\{v_i, v_j\}$, $\{v_p, v_q\}$ with $i < j, p < q$, it holds $\chi(\{v_i,v_j\}) = \chi(\{v_p,v_q\})$ whenever $i = p$. Equivalently, each color class among the edges is determined solely by the lower-indexed endpoint in the ordering. A rainbow clique of size $t$ is a $K_t$ subgraph in which all edges have distinct colors.

The unordered canonical Ramsey number $CR(s,t)$ is the smallest $n$ such that every coloring $\chi$ of $K_n$ contains either an orderable $K_s$ or a rainbow $K_t$ (Brosch et al., 6 Nov 2025). If $n < CR(s,t)$, there exists a coloring that avoids both types of subgraphs.

2. Exact Values for Small Parameters

Substantial computational effort elucidates the spectrum of $CR(s,t)$ for small $s, t$. Using a blend of integer programming (ILP), heuristics, exhaustive search, and flag algebra infeasibility certificates, the following exact results are established (Brosch et al., 6 Nov 2025):

$(s, t)$	$CR(s, t)$	Methods Used
$(3, 3)$	$3$	Trivial, direct analysis
$(3, 4)$	$7$	ILP, enumeration
$(4, 3)$	$6$	ILP, enumeration
$(5, 3)$	$11$	Tabu search, ILP
$(3, 5)$	$13$	Tabu search, flag algebra
$(6, 3)$	$26$	Enumeration, flag algebra

All other pairs $(s, t)$ with $3\le s\le6$ and $3\le t\le5$ remain unresolved.

Proof sketch for $CR(3,5)=13$: The lower bound is constructed by tabu search producing a $K_{12}$ coloring with no orderable triangle and no rainbow $K_5$, whereas the upper bound employs flag algebra calculations certifying that any coloring of $K_{13}$ must yield one of the two forbidden substructures. For $CR(6,3)=26$, exhaustive isomorphism-reduced enumeration proves the lower bound, while flag algebra SDPs on six-vertex patterns give the upper bound.

3. General Bounds and Asymptotics

Richer's original bounds prescribe, for all $s, t \ge 3$ (Brosch et al., 6 Nov 2025):

$$\left( \binom{t}{2} - 1 \right)^{s-2} + 1 \leq CR(s, t) \leq 7^{3-s} t^{4s-4}$$

Recent probabilistic analysis refines the asymptotic growth to (Araujo et al., 17 Sep 2024):

$$CR(s,t) = \Theta\left( \frac{t^3}{\log t} \right)^{s-2},\quad \text{as } t\to\infty, \ \text{with}\ s\ \text{fixed}$$

This matches the lower bound from Jiang’s blow-up anti-Ramsey construction and surpasses previous results that had the exponent $s-1$ instead of $s-2$ (Araujo et al., 17 Sep 2024). For fixed $s$ and increasing $t$, the unordered canonical Ramsey numbers exhibit polynomial growth rather than the doubly exponential behavior seen in ordered canonical Ramsey numbers.

4. Computational and Analytic Methodologies

Lower bounds ($CR(s,t) \geq n+1$) typically arise via explicit constructions of colorings lacking both orderable $K_s$ and rainbow $K_t$. Three principal techniques are invoked (Brosch et al., 6 Nov 2025):

• Tabu Search: Random initialization followed by iterative edge-recoloring to minimize the sum of forbidden structures, accepting non-local moves to prevent cycling.
• Integer Linear Programming (ILP):
  • Color-explicit formulation: one variable per edge-color pairing, with constraints eliminating forbidden cliques.
  • Colorblind formulation: variables indicate which edges share colors, accompanied by triangle inequalities to enforce transitivity.
• Enumeration: Exhaustive enumeration up to isomorphism for small $n$, notably in settling $CR(6,3)$.

Upper bounds ($CR(s, t) \leq N$) are established via flag algebra methods, particularly in the blow-up framework (Brosch et al., 6 Nov 2025). Balancing subgraph densities and forbidden triple patterns yields a semidefinite program; infeasibility is certified when maximum non-edge density falls below $1/N$.

The colorblind flag algebra implementation of Lidický–Pfender, extended for ordered and color partition settings, is commonly employed. Large-scale SDPs are solved using CSDP or MOSEK on high-capacity computational clusters.

5. Off-Diagonal and Variants

The unordered canonical Ramsey paradigm extends naturally to more general cases $CR(G, H)$ for arbitrary graphs $G, H$, as well as the ER$(m, \ell, r)$ variant where the goal is to ensure monochromatic, lexical, or rainbow copies of prescribed order (Araujo et al., 17 Sep 2024). Results for ER$(m, \ell, r)$ are as follows:

• $ER(m,3,r) \leq C_1 m \frac{r^3}{\log r}$
• $$ER(3,\ell,r) \leq C_2 \left( \frac{r^3}{\log r} \right)^{\ell-2}$$
• $ER(4,4,r) \leq C_3 \frac{r^4}{\log r}$

These bounds are all tight up to constants, via adaptations of probabilistic, blow-up, and deletion techniques. For bipartite graphs and degenerate trees, polynomial upper bounds supplanted earlier exponential results (see also (Gishboliner et al., 11 Oct 2024)).

6. Open Problems and Directions

Several salient open problems persist (Brosch et al., 6 Nov 2025):

• Determining $CR(4,4)$, presently bounded by $26 \leq CR(4,4)\leq 45$.
• Generalizing off-diagonal cases $CR(G,H)$: current bounds for bipartite $G$ are wide, with many instances unresolved.
• Analyzing asymptotic behavior for $CR(s,t)$ as $s,t\to\infty$ in refined regimes.
• Extending methodologies to hypergraphs, directed graphs, and ordered variants.

A plausible implication is that further development of flag algebra machinery and probabilistic constructions will refine these bounds and open paths to new extremal and structural phenomena.

7. Connections and Comparisons

Unordered canonical Ramsey numbers display marked differences from both classical Ramsey numbers $R(k)$ and ordered Erdős–Rado numbers $\operatorname{er}(t)$ (Araujo et al., 17 Sep 2024). Classical $R(k)$ grows as $2^{\Theta(k)}$, while ordered canonical numbers are doubly exponential, $\operatorname{er}(t) \approx 2^{\Theta(t^2)}$. In contrast, unordered canonical Ramsey numbers for fixed $s$ grow only polynomially in $t$ (modulo logarithmic factors).

They are also closely linked to anti-Ramsey results, with Babai’s bounds for rainbow cliques serving as a foundational component in current proofs. The effect of permitting arbitrary coloring but seeking canonical substructure yields a dramatic reduction in necessary $n$ compared to monochromatic or rainbow-only variants.

The combination of computational heuristics and analytic flag algebra frameworks currently yields the sharpest known bounds, and these approaches are expected to drive future progress in the domain.