Ramsey Goodness of Paths

• Ramsey goodness of paths is defined by the equality R(Pₙ, H) = (n-1)(χ(H)-1) + σ(H), linking path lengths to chromatic and partition parameters.
• Research shows that for any graph H, large n ensures Pₙ is H-good, with refined bounds in unbalanced multipartite cases and for bounded-degree trees using combinatorial techniques.
• Extensions to hypergraphs reveal distinct behavior where loose paths regain asymptotic goodness while tighter configurations face additional bounds, underscoring nuanced threshold phenomena.

A path is said to be Ramsey-good with respect to a graph or hypergraph $H$ if the Ramsey number $R(P_n,H)$ coincides with the natural lower bound predicted by chromatic and partition parameters. The mathematical study of this property—the Ramsey goodness of paths—plays a central role in structural Ramsey theory for both graphs and hypergraphs, with deep links to extremal combinatorics, structural embeddings, probabilistic constructions, and the study of graph parameters such as bandwidth and degree constraints.

1. Foundational Definitions and Lower Bounds

Let %%%%2%%%% and $H$ be graphs. The Ramsey number $R(G,H)$ is the smallest integer $N$ such that any red/blue coloring of the edges of $K_N$ contains either a red copy of $G$ or a blue copy of $H$. If $G$ is connected, the fundamental lower bound (due to Burr) is

$$R(G,H) \geq (|G|-1)\bigl(\chi(H)-1\bigr) + \sigma(H),$$

where $\chi(H)$ is the chromatic number of $H$, and $\sigma(H)$ is the size of the smallest color class in any optimal $\chi(H)$-coloring of $H$ (Pokrovskiy et al., 2015). A graph $G$ is termed $H$-good if equality holds.

Applied to paths $P_n$, the Ramsey-goodness problem asks for which $H$ and $n$ one has

$R(P_n,H) = (n-1)\bigl(\chi(H)-1\bigr)+\sigma(H).$

For $k$-uniform hypergraphs, the analogous lower bound is

$R(G, H) \geq (v(G)-1)\cdot (\chi(H)-1) + \sigma(H)$

where $v(G)$ is the order of $G$ and $\chi(H), \sigma(H)$ are defined for hypergraphs as in the graph case (Boyadzhiyska et al., 2023).

2. Ramsey Goodness of Paths in Graphs

The principle result established by Pokrovskiy and Sudakov is that for any graph $H$, the $n$-vertex path $P_n$ is $H$-good for all $n \ge 4|H|$:

$$R(P_n, H) = (n-1)\bigl(\chi(H)-1\bigr) + \sigma(H), \qquad \text{for } n \geq 4|H| \, [1512.07874, 2410.19942],$$

where $|H|$ is the order of $H$ (Botler et al., 25 Oct 2024). The proof relies on inductive embedding techniques, the use of the Pósa rotation-extension lemma, and structural partitioning via color classes, showing that for large enough $n$, the extremal construction—the union of appropriately sized red cliques with all other edges colored blue—provides the only obstruction.

Botler, Moreira, and de Souza refined this to $(2+\varepsilon)|H|$ in the highly unbalanced multipartite case, under the additional constraint that the part sizes $m_1,\ldots,m_k$ satisfy a quadratic unbalance condition:

$$\varepsilon m_i \geq 2 m_{i-1}^2 \quad (2 \le i \le k),$$

yielding

$$R(P_n, K_{m_1,\ldots,m_k}) = (n-1)(k-1)+m_1, \quad n \ge (2+\varepsilon)(m_1+\ldots+m_k) \, [2410.19942].$$

The achieved improvement is sharp up to a constant factor: no bound below $2|H|$ can hold for all $H$ (Botler et al., 25 Oct 2024).

For bounded-degree trees, Balla, Pokrovskiy, and Sudakov demonstrated that for any fixed $\Delta$, every $n$-vertex tree $T$ of maximum degree at most $\Delta$ is $H$-good for $n\ge C(\Delta)|H|\log^4|H|$ (Balla et al., 2016).

The table below summarizes key established Ramsey-goodness ranges for $P_n$ vs. $H$:

$H$	Minimal $n$	Bound	Source
Arbitrary $H$	$n \ge 4|H|$	Equality	(Pokrovskiy et al., 2015)
Unbalanced $H$	$n \ge (2+\varepsilon)|H|$	Equality	(Botler et al., 25 Oct 2024)
Trees	$n \ge C|H|\log^4|H|$	Asymptotic	(Balla et al., 2016)

3. Degree and Structural Conditions in Host Graphs

Beyond complete graphs as hosts, recent research has elucidated tight minimum degree conditions for dense, but possibly incomplete, host graphs to ensure Ramsey-goodness for paths. Aragão, Marciano, and Mendonça proved that if $G$ is an $N\ge (n-1)(m-1)+1$ vertex graph with $\delta(G)\geq N-\lceil n/2\rceil$, then $G \to (P_n, K_m)$ (Aragão et al., 20 Mar 2024, Luo et al., 4 Dec 2025). Luo and Peng further improved the threshold for arbitrary trees and, in the case of non-star trees (including all $P_n$ with $n\ge 3$), reduced the minimum degree sufficient for Ramsey-goodness even further (Luo et al., 4 Dec 2025).

For arbitrary pairs $(K_r, P_t)$, Aragão, Marciano, and Mendonça established the sharp minimum degree threshold:

$$\delta(G)\geq n-\lceil t/2\rceil, \quad n=(r-1)(t-1)+1$$

is both necessary and sufficient for $G\to(K_r, P_t)$ (Aragão et al., 20 Mar 2024).

4. Ramsey Goodness for Sparse or Random Hosts

Fan and Lin proved that for sparse connected graphs $G$ on $n \ge \Omega(k^4)$ vertices with $e(G)\le (1+1/(ck^2))n$ (for some constant $c$), $G$ is $P_k$-good:

$$r(G, P_k) = \max\{ n+\lfloor k/2\rfloor -1,\ n + k -2 - \alpha'(G) - \gamma \} \ [2507.11835],$$

bridging a forty-year gap with the previous $n\ge \Omega(k^{12})$ requirement (Fan et al., 16 Jul 2025).

For random graphs, Letzter and Sahasrabudhe determined sharp thresholds: for $r\ge2$,

$$G((1+\varepsilon)rn, p)\to (K_{r+1}, P_n),\qquad p\gg n^{-2/(r+1)} \ [1909.00030].$$

The thresholds depend delicately on both the order and density.

5. Asymptotic and Structural Ramifications

Allen, Brightwell, and Skokan proved that for any fixed $H$ and for all sufficiently large $n$, if $P_n$ (or, in general, any bounded degree, bounded bandwidth graph) is taken as host, then path-goodness always holds:

$$R(P_n, H) = (n-1)\bigl(\chi(H)-1\bigr) + \sigma(H) \ [1010.5079].$$

This applies also to any $G$ with $\Delta(G)\leq \Delta$ and bandwidth $o(n)$.

When $G_n$ is a bounded-degree graph of order $n$ with $\alpha(G_n)\leq n/4$, then $P_n$ is asymptotically $G_n$-good:

$$R(P_n, G_n) = (k-1)(n-1) + \sigma(G_n) + o(n) \ [1407.7092].$$

6. Ramsey Goodness of Paths in Hypergraphs

Ramsey-goodness phenomena diverge profoundly for $k$-uniform hypergraphs. For $k$-uniform $\ell$-paths ($\ell\ge2$), one generally has failure of goodness for a large class of $k$-graphs $H$, with additional terms in the lower bound preventing equality; e.g.,

$$R(P^{(k)}_{n, \ell}, H) \ge (\chi(H)-1)(n-1) + \lfloor n/k \rfloor > (\chi(H)-1)(n-1) + \sigma(H)$$

for many $H$ (Boyadzhiyska et al., 2023).

By contrast, for loose paths ($\ell=1$), asymptotic Ramsey-goodness is restored:

$R(P^{(k)}_{n,1}, H) = (\chi(H)-1)n + O(1)$

as $n\to\infty$ (Boyadzhiyska et al., 2023).

In the 3-uniform setting, tight paths are $\mathbb F$-good for the Fano plane $\mathbb F$, with

$R(P_n, \mathbb F) = 2n-1 \ [1901.07097].$

Methods involve combinatorial decompositions into red clique "blobs," butterfly structures, Turán-type arguments on auxiliary graphs, and path-decomposition followed by interlaced embeddings.

7. Open Problems and Further Directions

Key open directions include:

• Determining the minimal constant $c$ such that $P_n$ is $H$-good for $n > c|H|$ for all $H$.
• Eliminating the logarithmic factor in $n$ for bounded-degree tree-goodness results (Balla et al., 2016).
• Closing the remaining gaps in minimum degree conditions for dense (not complete) host graphs (Luo et al., 4 Dec 2025).
• Extending goodness results from paths to cycles and graphs of small bandwidth or bounded treewidth (Botler et al., 25 Oct 2024, Allen et al., 2010).
• Understanding precise thresholds and obstructions for Ramsey-goodness in nontrivial hypergraph pairs (Boyadzhiyska et al., 2023, Balogh et al., 2019).

Further, for random hosts, exact threshold functions for the Ramsey property involving paths remain an area of active study, particularly the interplay of size and edge probability (Moreira, 2019).

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The field of Ramsey goodness for paths illustrates the interplay of extremal constructions, probabilistic methods, and deep structure theory at the interface of graph Ramsey theory. The comprehensive results for graphs contrast sharply with the much more delicate and nuanced landscape in uniform hypergraphs, where goodness can fail dramatically except in the asymptotic sense for certain path types or highly structured target hypergraphs.