---
title: Fan-Goodness in Ramsey Theory
url: https://www.emergentmind.com/topics/fan-goodness
type: topic
---

# Fan-Goodness in Ramsey Theory

Searching arXiv for the cited papers and closely related work on fan-goodness in Ramsey theory.
Fan-goodness is a Ramsey-theoretic notion attached to the fan graph \(F_m=K_1+mK_2\), equivalently a graph of \(m\) triangles sharing a common vertex. For graphs \(G\) and \(H\), the Ramsey number \(r(G,H)\) is the least \(N\) such that every red–blue edge-coloring of \(K_N\) contains a red copy of \(G\) or a blue copy of \(H\). In the classical orientation, one says that \(H\) is \(G\)-good when the Burr lower bound is attained with equality; in the fan setting this yields statements such as \(F_n\) being \(K_p\)-good, i.e. \(r(K_p,F_n)=2(p-1)n+1\). In a dual orientation used for sparse-source problems, a connected \(n\)-vertex graph \(G\) is called fan-good if \(r(G,F_k)=2n-1\). Recent work has turned fan-goodness from a phenomenon previously accessible only through tower-type bounds into a quantitatively explicit theory with polynomial and quadratic thresholds, generalized-fan extensions, and sparse-graph criteria [2208.05829][2310.13204][2507.09832].

## 1. Definitions, lower bounds, and notational conventions

Given graphs \(G\) and \(H\), Burr’s lower bound states
\[
r(G,H)\ge (\chi(G)-1)(|H|-1)+s(G),
\]
where \(\chi(G)\) is the chromatic number of \(G\) and \(s(G)\) is the chromatic surplus, namely the minimum size of any color-class in a \(\chi(G)\)-coloring of \(G\). When equality holds, \(H\) is \(G\)-good. In particular, for \(G=K_p\) one has \(\chi(G)=p\) and \(s(G)=1\), so
\[
r(K_p,H)\ge (p-1)(|H|-1)+1.
\]
Thus \(H\) is \(K_p\)-good exactly when
\[
r(K_p,H)=(p-1)(|H|-1)+1.
\]
This is the formulation used in the fan-complete and generalized-fan literature [2208.05829].

The fan graph is
\[
F_m=K_1+mK_2,
\]
where \(+\) denotes the join and \(mK_2\) denotes the union of \(m\) disjoint copies of \(K_2\). Its order is \(1+2m\). More generally, a generalized fan is a graph of the form
\[
K_1+nH,
\]
obtained from \(n\) disjoint copies of a fixed graph \(H\) together with an additional hub joined to every vertex of \(nH\) [2208.05829][2310.13204].

A second orientation, used in recent sparse-graph work, fixes the fan as the target graph. Since \(F_k\) has \(\chi(F_k)=3\) and \(s(F_k)=1\), Burr’s bound becomes
\[
r(G,F_k)\ge 2|V(G)|-1.
\]
For a connected graph \(G\) of order \(n\), fan-goodness in this sense means
\[
r(G,F_k)=2n-1.
\]
Likewise, if \(tF_k\) is the disjoint union of \(t\) copies of \(F_k\), then \(\chi(tF_k)=3\) and \(s(tF_k)=t\), so
\[
r(G,tF_k)\ge 2n+t-2,
\]
and \(G\) is \(tF_k\)-good when equality holds [2507.09832].

## 2. Ordinary fans as \(K_p\)-good graphs

The first explicit polynomial threshold for ordinary fan-goodness in the \(K_p\)-good direction was established by Chung and Lin. For \(p\ge 3\), if
\[
n\ge 27p^2,
\]
then
\[
r(K_p,F_n)=(p-1)(|F_n|-1)+1=2(p-1)n+1.
\]
Equivalently, \(F_n\) is \(K_p\)-good for all such \(n\). The same work records a slightly stronger corollary: if
\[
n>\frac{(3+3\sqrt2)^2}{2}\,p^2\approx 26.228\,p^2,
\]
then \(F_n\) is \(K_p\)-good. These results improve earlier tower-type lower bounds for \(n\) due to Li and Rousseau (1996) [2208.05829].

This development places fans within the broader theory of Ramsey-goodness. Classical examples include the theorem that all trees \(T_n\) satisfy
\[
r(K_p,T_n)=(p-1)(n-1)+1,
\]
so all trees are \(K_p\)-good. The fan family is structurally sparser than general dense targets and, as noted in the literature, interpolates between stars and more complex sparse graphs. The fan-goodness problem therefore became a test case for replacing regularity-based existence arguments by quantitatively controlled stability and supersaturation arguments [2208.05829].

## 3. Generalized fans \(K_1+nH\) and the collapse of the universal constant

For a fixed graph \(H\) of order \(h\), let \(\ell=r(K_p,H)\). Chung and Lin proved that the generalized fan \(K_1+nH\) is \(p\)-good provided
\[
n\ge c_0\frac{p\ell}{h},
\qquad
c_0=(3+3\sqrt2)^2\approx 52.456.
\]
Subsequent work reduced this constant in three stages: first to \(3\) via the Andrásfai–Erdős–Sós theorem, then to \(2\) via a Chen–Zhang-style structural decomposition, and finally to \(1\) through a new neighbor-counting and induction argument. Thus \(K_1+nH\) is \(p\)-good as soon as
\[
n\ge \frac{p\ell}{h}.
\]
In the special case \(H=K_2\), where \(\ell=r(K_p,K_2)=p\), the same paper obtained the refined fan bound
\[
r(K_p,F_n)=2(p-1)n+1
\quad\text{whenever}\quad
n\ge \frac{p^2-p-2}{2},
\]
for every \(p\ge 3\) [2310.13204].

The progression of sufficient conditions is summarized below.

| Setting | Bound on \(n\) | Consequence |
|---|---:|---|
| Generalized fan \(K_1+nH\) | \(n\ge c_0\,p\ell/h\), \(c_0\approx 52.456\) | \(K_1+nH\) is \(p\)-good |
| Generalized fan \(K_1+nH\) | \(n\ge 3\,p\ell/h\) | \(K_1+nH\) is \(p\)-good |
| Generalized fan \(K_1+nH\) | \(n\ge 2\,p\ell/h\) | \(K_1+nH\) is \(p\)-good |
| Generalized fan \(K_1+nH\) | \(n\ge p\ell/h\) | \(K_1+nH\) is \(p\)-good |
| Ordinary fan \(F_n\) | \(n\ge (p^2-p-2)/2\) | \(r(K_p,F_n)=2(p-1)n+1\) |

These results answer the quantitative question of how small \(n\) can be while preserving \(p\)-goodness of \(K_1+nH\). A plausible implication is that generalized-fan goodness is unusually responsive to fine structural control of near-extremal graphs, since the universal constant falls from approximately \(52.456\) to \(1\) without changing the overall template \(n\asymp p\ell/h\) [2310.13204].

## 4. Multipartite refinements and the discrepancy from \(G\)-goodness

Chung and Lin also treated the case where the source graph is a complete \(p\)-partite graph
\[
G=K_p(1,a_2,\dots,a_p),
\qquad
1=a_1<a_2\le \cdots \le a_p.
\]
For any fixed graph \(H\) with \(|H|=h\), under mild conditions on the \(a_i\) and for \(n\) sufficiently large, they proved the exact piecewise formula
\[
r(G,K_1+nH)=
\begin{cases}
(p-1)(nh+a_2-1)+1 & \text{if \(nh+a_2-1\) is even or \(a_2-1\) is even,}\\[4pt]
(p-1)(nh+a_2-2)+1 & \text{otherwise.}
\end{cases}
\]
This is a strengthened lower bound inequality for Ramsey numbers of the form \(r(G,K_1+F)\), specialized to multipartite \(G\) with one singleton part [2208.05829].

Two specializations are singled out in the literature. When \(H=K_2\), the formula recovers the fan setting. When \(H=K_1\), it gives an exact formula for \(r(G,K_{1,n})\) for sufficiently large \(n\), thereby answering Burr’s question from 1981 about the discrepancy of \(r(G,K_{1,n})\) from the naive \(G\)-good prediction. The parity split in the formula shows that the deviation from goodness is not arbitrary: it is controlled by the parity of \(nh+a_2-1\) and \(a_2-1\), with the two cases differing by exactly \(p-1\) in the main linear term [2208.05829].

## 5. Fan-goodness of sparse graphs and multiple fans

A distinct but related line of work fixes the target as a fan and asks which connected sparse graphs are \(F_k\)-good. Let \(G\) be a connected graph of order \(n\), and let \(F_k=K_1+kK_2\). Brennan had shown that \(r(G,F_k)=2n-1\) for unicyclic \(G\) when \(n\ge k^2-k+1\) and \(k\ge 18\), and asked for a threshold \(c(n)\) such that \(r(G,F_k)\ge 2n\) holds for graphs containing at least \(c(n)\) cycles. The sparse-graph theory answers this in a linear-density regime [2507.09832].

For fixed \(k\), define
\[
\epsilon(k)=\frac1{204k^3+126k^2}.
\]
If \(G\) is connected on \(n\) vertices with
\[
e(G)\le n(1+\epsilon(k))
\quad\text{and}\quad
n\ge 36k^4,
\]
then
\[
r(G,F_k)=2n-1.
\]
Thus every such \(G\) is fan-good. More generally, for fixed \(k,t\), define
\[
\epsilon(k,t)=\frac1{204tk^3+147tk^2}.
\]
If \(G\) is connected on \(n\) vertices with
\[
e(G)\le n(1+\epsilon(k,t))
\quad\text{and}\quad
n\ge 161t^2k^4,
\]
then
\[
r(G,tF_k)=2n+t-2.
\]
Hence the same density principle extends to the disjoint union of \(t\) fans [2507.09832].

These theorems subsume several earlier positive results for sparse classes. Trees and stars satisfy
\[
r(T_n,F_k)=2n-1 \quad \text{for } n\ge 3k^2-2k-1,
\]
and
\[
r(K_{1,n-1},F_k)=2n-1 \quad \text{for } n\ge k^2-k+1.
\]
Unicyclic graphs \(UC_n\) satisfy
\[
r(UC_n,F_k)=2n-1
\quad\text{for } k\ge 18 \text{ and } n\ge k^2-k+1.
\]
In Brennan’s terminology, the new results imply that the threshold \(c(n)\) for the number of cycles beyond which \(r(G,F_k)\ge 2n\) is greater than \(\epsilon(k)n\). The stated theorems therefore identify a concrete \(n+o(n)\)-edge regime in which connected graphs are forced to be fan-good [2507.09832].

## 6. Proof architecture, quantitative shift, and terminological boundaries

The modern proofs of fan-goodness are notable for the replacement of regularity-based arguments by explicit structural and counting tools. In Chung–Lin’s treatment of \(F_n\) and \(K_1+nH\), the key ingredients are a degree-majorization argument, counting \(K_p\)’s and supersaturation, the Fox–He–Wigderson stability-supersaturation lemma, and the extraction of large independent sets. The stability-supersaturation input replaces the Szemerédi regularity lemma entirely and yields the precise near-Turán partition structure needed for the argument. By careful tracking of constants, the resulting bounds on \(n\) are polynomial rather than tower-type [2208.05829].

The later paper on generalized fans organizes the improvement of the universal constant into three proof paradigms. The \(c=3\) bound uses the Andrásfai–Erdős–Sós theorem and a minimum-degree estimate to force a \((p-1)\)-partite structure. The \(c=2\) bound uses a Chen–Zhang-style partition \(U_1,\dots,U_p\) together with double counting and further partitioning of the exceptional set. The \(c=1\) bound uses a new two-stage common-neighbor argument combined with induction on \(p\). In the fan case \(H=K_2\), these methods yield the explicit threshold \((p^2-p-2)/2\) [2310.13204].

The sparse-graph theory uses a different toolkit: a trichotomy lemma for sparse graphs, the Bondy–Erdős path-extension lemma, Hall’s marriage theorem, a matching-Ramsey lemma of Faudree–Schelp–Sheehan, and a new inductive upper bound
\[
r(G,F_k)\le n+2k\,e(G)-\frac{2e(G)}n.
\]
The authors note an inherent trade-off between the lower bound on \(n\) and the upper bound on \(e(G)\) in their trichotomy argument, and state that substantially improving these bounds would require new ideas. Natural open problems include determining the minimal growth rate of \(e(G)\), or of the number of cycles, that spoils \(F_k\)-goodness; sharpening the dependence of \(n\) on \(k\); and extending fan-goodness to other classes of sparse graphs or other target graphs beyond fans [2507.09832].

The term should be distinguished from two unrelated graph-theoretic notions that involve the name “Fan” or the word “good.” First, the Fan-type degree-sum condition for completely independent spanning trees states that if a connected graph \(G\) of order \(n\ge 7\) satisfies \(\mu_2(G)\ge n\), then \(G\) contains two completely independent spanning trees; this concerns CIST existence, not Ramsey goodness [2502.11522]. Second, the graph-labeling literature studies edge-graceful usual fan graphs \(F_{1,n}\), where \(F_{1,n}\) is formed from a hub joined to a path \(P_n\); in that setting \(F_{1,n}\) is edge-graceful exactly when \(n\in\{2,3,11\}\), which is a different construction and a different notion of “goodness” [2412.08338].

Source: https://www.emergentmind.com/topics/fan-goodness