Fan-Goodness in Ramsey Theory

• Fan-goodness is a Ramsey-theoretic property defining conditions where fan graphs—formed by triangles sharing a common vertex—attain exact Ramsey number bounds.
• It establishes explicit thresholds for both ordinary and generalized fan configurations, reducing previous tower-type bounds to polynomial and quadratic thresholds.
• It extends to sparse graphs and multipartite refinements, highlighting practical combinatorial strategies that balance edge conditions and structural control.

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 $m$0-good when the Burr lower bound is attained with equality; in the fan setting this yields statements such as $m$1 being $m$2-good, i.e. $m$3. In a dual orientation used for sparse-source problems, a connected $m$4-vertex graph $m$5 is called fan-good if $m$6. 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 (Chung et al., 2022, Zhang et al., 2023, Huang et al., 13 Jul 2025).

1. Definitions, lower bounds, and notational conventions

Given graphs $m$7 and $m$8, Burr’s lower bound states

$m$9

where $G$0 is the chromatic number of $G$1 and $G$2 is the chromatic surplus, namely the minimum size of any color-class in a $G$3-coloring of $G$4. When equality holds, $G$5 is $G$6-good. In particular, for $G$7 one has $G$8 and $G$9, so

$H$0

Thus $H$1 is $H$2-good exactly when

$H$3

This is the formulation used in the fan-complete and generalized-fan literature (Chung et al., 2022).

The fan graph is

$H$4

where $H$5 denotes the join and $H$6 denotes the union of $H$7 disjoint copies of $H$8. Its order is $H$9. More generally, a generalized fan is a graph of the form

$r(G,H)$0

obtained from $r(G,H)$1 disjoint copies of a fixed graph $r(G,H)$2 together with an additional hub joined to every vertex of $r(G,H)$3 (Chung et al., 2022, Zhang et al., 2023).

A second orientation, used in recent sparse-graph work, fixes the fan as the target graph. Since $r(G,H)$4 has $r(G,H)$5 and $r(G,H)$6, Burr’s bound becomes

$r(G,H)$7

For a connected graph $r(G,H)$8 of order $r(G,H)$9, fan-goodness in this sense means

$N$0

Likewise, if $N$1 is the disjoint union of $N$2 copies of $N$3, then $N$4 and $N$5, so

$N$6

and $N$7 is $N$8-good when equality holds (Huang et al., 13 Jul 2025).

2. Ordinary fans as $N$9-good graphs

The first explicit polynomial threshold for ordinary fan-goodness in the $K_N$0-good direction was established by Chung and Lin. For $K_N$1, if

$K_N$2

then

$K_N$3

Equivalently, $K_N$4 is $K_N$5-good for all such $K_N$6. The same work records a slightly stronger corollary: if

$K_N$7

then $K_N$8 is $K_N$9-good. These results improve earlier tower-type lower bounds for $G$0 due to Li and Rousseau (1996) (Chung et al., 2022).

This development places fans within the broader theory of Ramsey-goodness. Classical examples include the theorem that all trees $G$1 satisfy

$G$2

so all trees are $G$3-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 (Chung et al., 2022).

3. Generalized fans $G$4 and the collapse of the universal constant

For a fixed graph $G$5 of order $G$6, let $G$7. Chung and Lin proved that the generalized fan $G$8 is $G$9-good provided

$H$0

Subsequent work reduced this constant in three stages: first to $H$1 via the Andrásfai–Erdős–Sós theorem, then to $H$2 via a Chen–Zhang-style structural decomposition, and finally to $H$3 through a new neighbor-counting and induction argument. Thus $H$4 is $H$5-good as soon as

$H$6

In the special case $H$7, where $H$8, the same paper obtained the refined fan bound

$H$9

for every $H$0 (Zhang et al., 2023).

The progression of sufficient conditions is summarized below.

Setting	Bound on $H$1	Consequence
Generalized fan $H$2	$H$3, $H$4	$H$5 is $H$6-good
Generalized fan $H$7	$H$8	$H$9 is $m$00-good
Generalized fan $m$01	$m$02	$m$03 is $m$04-good
Generalized fan $m$05	$m$06	$m$07 is $m$08-good
Ordinary fan $m$09	$m$10	$m$11

These results answer the quantitative question of how small $m$12 can be while preserving $m$13-goodness of $m$14. 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 $m$15 to $m$16 without changing the overall template $m$17 (Zhang et al., 2023).

4. Multipartite refinements and the discrepancy from $m$18-goodness

Chung and Lin also treated the case where the source graph is a complete $m$19-partite graph

$m$20

For any fixed graph $m$21 with $m$22, under mild conditions on the $m$23 and for $m$24 sufficiently large, they proved the exact piecewise formula

$m$25

This is a strengthened lower bound inequality for Ramsey numbers of the form $m$26, specialized to multipartite $m$27 with one singleton part (Chung et al., 2022).

Two specializations are singled out in the literature. When $m$28, the formula recovers the fan setting. When $m$29, it gives an exact formula for $m$30 for sufficiently large $m$31, thereby answering Burr’s question from 1981 about the discrepancy of $m$32 from the naive $m$33-good prediction. The parity split in the formula shows that the deviation from goodness is not arbitrary: it is controlled by the parity of $m$34 and $m$35, with the two cases differing by exactly $m$36 in the main linear term (Chung et al., 2022).

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 $m$37-good. Let $m$38 be a connected graph of order $m$39, and let $m$40. Brennan had shown that $m$41 for unicyclic $m$42 when $m$43 and $m$44, and asked for a threshold $m$45 such that $m$46 holds for graphs containing at least $m$47 cycles. The sparse-graph theory answers this in a linear-density regime (Huang et al., 13 Jul 2025).

For fixed $m$48, define

$m$49

If $m$50 is connected on $m$51 vertices with

$m$52

then

$m$53

Thus every such $m$54 is fan-good. More generally, for fixed $m$55, define

$m$56

If $m$57 is connected on $m$58 vertices with

$m$59

then

$m$60

Hence the same density principle extends to the disjoint union of $m$61 fans (Huang et al., 13 Jul 2025).

These theorems subsume several earlier positive results for sparse classes. Trees and stars satisfy

$m$62

and

$m$63

Unicyclic graphs $m$64 satisfy

$m$65

In Brennan’s terminology, the new results imply that the threshold $m$66 for the number of cycles beyond which $m$67 is greater than $m$68. The stated theorems therefore identify a concrete $m$69-edge regime in which connected graphs are forced to be fan-good (Huang et al., 13 Jul 2025).

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 $m$70 and $m$71, the key ingredients are a degree-majorization argument, counting $m$72’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 $m$73 are polynomial rather than tower-type (Chung et al., 2022).

The later paper on generalized fans organizes the improvement of the universal constant into three proof paradigms. The $m$74 bound uses the Andrásfai–Erdős–Sós theorem and a minimum-degree estimate to force a $m$75-partite structure. The $m$76 bound uses a Chen–Zhang-style partition $m$77 together with double counting and further partitioning of the exceptional set. The $m$78 bound uses a new two-stage common-neighbor argument combined with induction on $m$79. In the fan case $m$80, these methods yield the explicit threshold $m$81 (Zhang et al., 2023).

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

$m$82

The authors note an inherent trade-off between the lower bound on $m$83 and the upper bound on $m$84 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 $m$85, or of the number of cycles, that spoils $m$86-goodness; sharpening the dependence of $m$87 on $m$88; and extending fan-goodness to other classes of sparse graphs or other target graphs beyond fans (Huang et al., 13 Jul 2025).

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 $m$89 of order $m$90 satisfies $m$91, then $m$92 contains two completely independent spanning trees; this concerns CIST existence, not Ramsey goodness (Ma et al., 17 Feb 2025). Second, the graph-labeling literature studies edge-graceful usual fan graphs $m$93, where $m$94 is formed from a hub joined to a path $m$95; in that setting $m$96 is edge-graceful exactly when $m$97, which is a different construction and a different notion of “goodness” (Angel et al., 2024).