Generalised Ramsey Numbers Overview

• Generalised Ramsey numbers are extremal functions that determine the minimum conditions for the emergence of ordered structures under specific coloring constraints.
• They unify classical Ramsey theory and Turán-type problems by incorporating edge-coloring, multicolor, and hypergraph variants to analyze thresholds and structure formation.
• Research employs probabilistic, combinatorial, and algorithmic techniques—such as container methods and hypergraph matchings—to establish both polynomial and subpolynomial bounds.

A generalized Ramsey number is an extremal function encoding the minimal conditions under which a certain ordered structure must emerge, given constraints on coloring or forbidden subgraphs. This framework unifies and greatly extends classical Ramsey theory, connecting it with extremal combinatorics, Turán-type problems, coloring thresholds, structural graph theory, and even algorithmic and quantum methods.

1. Definitions and Main Variants

Generalized Ramsey numbers encompass several major classes, each parameterized according to the structural requirements imposed on subgraphs, or the palette and rules of coloring:

• Edge-coloring variant (Erdős–Shelah–Gyárfás function): For fixed $p\geq 2$ and $2\leq q\leq \binom{p}{2}$, a $(p,q)$-coloring of $K_n$ is an edge-coloring such that every $K_p$ uses at least $q$ colors. The minimum number of colors required is $f(n, p, q)$ (Conlon et al., 2014).
• Graph-pair Ramsey numbers: $R(G, H)$ is the least $N$ so that any red/blue edge-coloring of $K_N$ yields a red $G$ or blue $H$ (Brinkmann et al., 2012).
• Multicolor and set Ramsey numbers: For families of graphs $\Gamma_1, \Gamma_2$, $R(\Gamma_1, \Gamma_2)$ is the minimal $N$ such that any red/blue coloring of $K_N$ has a red member of $\Gamma_1$ or a blue member of $\Gamma_2$ (Hansson, 2016).
• Restricted-color Ramsey numbers: For a host $G$ and forbidden $H$, let $r(G,H,q)$ be the minimum $t$ so that every $t$-coloring of $E(G)$ makes every copy of $H$ span at least $q$ colors (Lane et al., 26 May 2024).
• Blowup Ramsey numbers: Given graphs $G$ and $H$, define $f_{G,H,r}(t)$ as the minimum $n$ so that $G[n]$ (the uniform $t$-blowup) is $r$-Ramsey for $H[t]$ (Souza, 2019).
• Gallai-Ramsey numbers and generalizations: Generalizations involving color-restricted partitions (e.g., $k$-Gallai colorings) with two-parameter thresholds (Magnant et al., 2019).

2. Principal Results and Thresholds

Generalized Ramsey numbers interpolate between classical Ramsey numbers and Turán-type extremal quantities. Key metatheorems include:

• Polynomial–subpolynomial dichotomy for $f(n,p,q)$: $f(n,p,q)$ has polynomial growth in $n$ for $q\geq p$, and is subpolynomial ($n^{o(1)}$) for $q\leq p-1$ (Conlon et al., 2014).

| $q$ value | Growth of $f(n,p,q)$ | |-------------------|-------------------------| | $q \leq p-1$ | $n^{o(1)}$ (subpoly.) | | $q \geq p$ | $n^{\Omega(1)}$ (poly.) |

• Linear and quadratic thresholds: Linear threshold is $q_{\mathrm{lin}}(p) = \binom{p}{2} - p+3$ and quadratic is $q_{\mathrm{quad}}(p) = \binom{p}{2} - p/2 + 2$. If $q = q_\mathrm{lin}(p)$ then $f(n,p,q)=\Theta(n)$; if $q=q_\mathrm{quad}(p)$ and $p$ even, $f(n,p,q)=\Theta(n^2)$ (Bennett et al., 2023, Bennett et al., 2 Aug 2024).
• Bounds on generalized Ramsey numbers for small graphs: Precise bounds and sometimes exact values—e.g., $f(n,5,8)=\frac67 n + o(n)$ (Bennett et al., 2 Aug 2024), $r(K_n, C_k, 3) = n/(k-2) + o(n)$ (Bal et al., 24 May 2024, Lane et al., 26 May 2024).
• Generalized Ramsey for cycles and paths: For cycles $C_k$ and 3-colorings, $f(K_n, C_k, 3)=n/(k-2)+o(n)$; for paths $P_\ell$, $$f(K_n, P_\ell, \lceil \ell/2\rceil + 1) = \Theta(n^2)$$ (Bal et al., 24 May 2024, Lane et al., 26 May 2024).

3. Proof Methods and Structural Techniques

Research on generalized Ramsey numbers employs a variety of combinatorial, probabilistic, and algorithmic techniques:

• Probabilistic local lemmas and container methods: Erdős–Gyárfás’ use of the Lovász Local Lemma (Conlon et al., 2014); container methods in the random graph blowup context (Souza, 2019).
• Forbidden submatching method: Recently, the forbidden submatching method has enabled not only polynomial but also logarithmic improvements in counting colorings across the “non-integral regime” of parameters, via the construction of auxiliary hypergraphs and perfect matchings therein (Bennett et al., 2022, Bennett et al., 2023).
• Conflict-free hypergraph/hypermatching method: State-of-the-art upper bounds, especially for cycle/path Ramsey numbers, rely on extracting large matchings in auxiliary hypergraphs, so that packed gadgets avoid dangerous substructures (Bal et al., 24 May 2024, Lane et al., 26 May 2024, Bennett et al., 2 Aug 2024).
• Extremal graph theory (Turán/Brown–Erdős–Sós connection): For quadratic threshold Ramsey numbers and related parameters, extremal problems on $r$-uniform hypergraphs (e.g., maximizing the number of edges in hypergraphs avoiding sets of vertices with too many edges) are leveraged to determine leading order terms (Bennett et al., 2023).
• Algebraic constructions: Linear and cyclic colorings, and recursive “template” constructions for building large graphs with prescribed clique-avoidance (Rowley, 2019).
• Quantum algorithms: Reformulation of Ramsey problems as energy minimization tasks for adiabatic quantum optimization—giving computational determination of classical generalized Ramsey numbers for small trees (Ranjbar et al., 2016).

4. Notable Particular Cases and Examples

Numerous concrete cases have been exactly determined or asymptotically pinned down:

Parameter	Value	Reference
$f(n,3,3)$	$n+O(1)$	(Bennett et al., 2023)
$f(n,4,5)$	$5n+o(n)$	(Bennett et al., 2023)
$f(n,5,8)$	$\frac{6}{7} n + o(n)$	(Bennett et al., 2 Aug 2024)
$r(K_n, C_k, 3)$	$\frac{n}{k-2} + o(n)$	(Bal et al., 24 May 2024)
$GR(3,K_4,2)$	$10$	(Lidický et al., 10 Jul 2024)
Blowup Ramsey, $K_2[n]\rightarrow K_3[t]$	$2^t < n < \exp(3.3\times 10^7 t)$	(Souza, 2019)
$f_{s,t}(n)$, Erdős–Rogers function	$f_{s,s+1}(n) = O(\sqrt{n}(\log n)^{O(1)})$	(Dudek et al., 2013)
Mixed star-stripe Ramsey	$R(S_{t_1},..., S_{t_s}, n_1K_2,...,n_cK_2)$: closed formula	(Omidi et al., 2017)

5. Ramsey Numbers for Sets, Color Patterns, and Critical Constructions

Generalizations of the Ramsey problem often include:

• Families of forbidden subgraphs: Generalizing to $R(\Gamma_1, \Gamma_2)$ for graph sets or cycles, yielding piecewise-linear closed forms when one family contains a short or even cycle (Hansson, 2016).
• Partition and coloring structure (Gallai-type): For $k$-Gallai colorings, the two-parameter generalized Gallai-Ramsey number $ggr_{k,\ell}(H)$ measures the minimal $N$ so that every $k$-Gallai $\ell$-coloring of $K_N$ contains a monochromatic $H$. This generalizes the exponential/linear threshold dichotomy in the number of colors ($\ell$) with constants and exponents depending on $k$ (Magnant et al., 2019).
• Critical graphs and extremal colorings: For various parameter regimes, the characterization of critical graphs (those just below the Ramsey threshold) reveals structural properties—often forcing bipartite, block, or partite configurations, as seen in the cycle families (Hansson, 2016).

6. Open Problems and Directions

Current research has sharpened the understanding of thresholds and the asymptotic order for many parameter combinations, but major open questions remain:

• Sharp constants: For many threshold functions (e.g., $f(n,p,q_{\mathrm{lin}}(p))$), the exact limiting ratios and their existence for all $p$ are not known (Bennett et al., 2023, Bennett et al., 2 Aug 2024).
• Non-integral regime lower bounds: A significant gap remains between upper and lower bounds, particularly for cases like $f(n,4,3)$, where the exponent is not tightly pinned down (Bennett et al., 2022).
• Extensions to hypergraphs: The precise order, constants, and structure in higher-uniformity analogues are mostly open beyond linear cases (Bal et al., 24 May 2024).
• Algorithmic and random graph thresholds: Whether the uniform exponential bound conjectured for blowup Ramsey holds universally, and for which host/target graphs, remains unresolved (Souza, 2019).
• List-coloring and local constraint variants: List assignment and locally-bounded colorings offer intriguing generalizations; log-factor improvements have been shown for some, but the limits of these methods demand further investigation (Bennett et al., 2022).

In sum, generalized Ramsey numbers serve as a nexus of extremal combinatorics, coloring, structural theory, and algorithmic (including quantum) perspectives. The area remains active, with major results on thresholds and bounds, but with many significant structural and computational questions open for future research.