Saturation Number of Ramsey-Minimal Families
Mike Ferrara
1
Jaehoon Kim
1 University
2 University
2
Elyse Yeager
2
of Colorado-Denver
of Illinois at Urbana-Champaign
yeager2@illinois.edu
MIGHTY
University of Detroit Mercy
29 March 2014
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
1 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallest
number of edges over all n-vertex graphs that are H-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallest
number of edges over all n-vertex graphs that are H-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallest
number of edges over all n-vertex graphs that are H-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallest
number of edges over all n-vertex graphs that are H-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallest
number of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F, a graph G is F-saturated if no
member of F is a subgraph of G , but for every e ∈ G , some member of F
is a subgraph of G + e.
The saturation number sat(n;F) is the smallest number of edges over
all n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
2 / 11
Ramsey-Minimal Families
Definitions
Given ”forbidden” graphs H1 , . . . , Hk , and any graph G , we write
G → (H1 , . . . , Hk ) if any k coloring of E (G ) contains a monochromatic
copy of Hi in color i, for some i.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
3 / 11
Ramsey-Minimal Families
Definitions
Given ”forbidden” graphs H1 , . . . , Hk , and any graph G , we write
G → (H1 , . . . , Hk ) if any k coloring of E (G ) contains a monochromatic
copy of Hi in color i, for some i.
Famous Example: K6 → (K3 , K3 ), but K5 6→ (K3 , K3 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
3 / 11
Ramsey-Minimal Families
Definitions
Given ”forbidden” graphs H1 , . . . , Hk , and any graph G , we write
G → (H1 , . . . , Hk ) if any k coloring of E (G ) contains a monochromatic
copy of Hi in color i, for some i.
Famous Example: K6 → (K3 , K3 ), but K5 6→ (K3 , K3 )
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
3 / 11
Ramsey-Minimal Families
Definitions
Given ”forbidden” graphs H1 , . . . , Hk , and any graph G , we write
G → (H1 , . . . , Hk ) if any k coloring of E (G ) contains a monochromatic
copy of Hi in color i, for some i.
Famous Example: K6 → (K3 , K3 ), but K5 6→ (K3 , K3 )
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Less Famous Example: K6 is (K3 , K3 )-Ramsey Minimal.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
3 / 11
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Less Famous Example: K6 is (K3 , K3 )-Ramsey Minimal.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
4 / 11
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Less Famous Example: K6 is (K3 , K3 )-Ramsey Minimal.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
4 / 11
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Less Famous Example: K6 is (K3 , K3 )-Ramsey Minimal.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
4 / 11
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Less Famous Example: K6 is (K3 , K3 )-Ramsey Minimal.
Definitions
Rmin (H1 , . . . , Hk ) = Rmin = {G : G is (H1 , . . . , Hk )-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
4 / 11
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Less Famous Example: K6 is (K3 , K3 )-Ramsey Minimal.
K6 ∈ Rmin (K3 , K3 )
Definitions
Rmin (H1 , . . . , Hk ) = Rmin = {G : G is (H1 , . . . , Hk )-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
4 / 11
Saturation of Ramsey-Minimal Families
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
5 / 11
Saturation of Ramsey-Minimal Families
Rmin (H1 , . . . , Hk ) Saturation
Suppose G is Rmin (H1 , . . . , Hk ) saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
5 / 11
Saturation of Ramsey-Minimal Families
Rmin (H1 , . . . , Hk ) Saturation
Suppose G is Rmin (H1 , . . . , Hk ) saturated.
G has no subgraph that is (H1 , . . . , Hk )-Ramsey minimal
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
5 / 11
Saturation of Ramsey-Minimal Families
Rmin (H1 , . . . , Hk ) Saturation
Suppose G is Rmin (H1 , . . . , Hk ) saturated.
G has no subgraph that is (H1 , . . . , Hk )-Ramsey minimal
Adding any edge to G creates a subgraph that is
(H1 , . . . , Hk )-Ramsey minimal
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
5 / 11
Saturation of Ramsey-Minimal Families
Rmin (H1 , . . . , Hk ) Saturation
Suppose G is Rmin (H1 , . . . , Hk ) saturated.
G has no subgraph that is (H1 , . . . , Hk )-Ramsey minimal
Adding any edge to G creates a subgraph that is
(H1 , . . . , Hk )-Ramsey minimal
I
For any e ∈ E (G ), G + e → (H1 , . . . , Hk )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
5 / 11
Saturation of Ramsey-Minimal Families
Rmin (H1 , . . . , Hk ) Saturation
Suppose G is Rmin (H1 , . . . , Hk ) saturated.
G has no subgraph that is (H1 , . . . , Hk )-Ramsey minimal
I
G 6→ (H1 , . . . , Hk )
Adding any edge to G creates a subgraph that is
(H1 , . . . , Hk )-Ramsey minimal
I
For any e ∈ E (G ), G + e → (H1 , . . . , Hk )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
5 / 11
Saturation of Ramsey-Minimal Families
Rmin (H1 , . . . , Hk ) Saturation
Suppose G is Rmin (H1 , . . . , Hk ) saturated.
G has no subgraph that is (H1 , . . . , Hk )-Ramsey minimal
I
G 6→ (H1 , . . . , Hk )
Pf: If G → (H1 , . . . , Hk ), we delete edges as long as the deletion does
not cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is
(H1 , . . . , Hk )-Ramsey minimal
I
For any e ∈ E (G ), G + e → (H1 , . . . , Hk )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
5 / 11
Saturation of Ramsey-Minimal Families
Rmin (H1 , . . . , Hk ) Saturation
Suppose G is Rmin (H1 , . . . , Hk ) saturated.
G has no subgraph that is (H1 , . . . , Hk )-Ramsey minimal
I
G 6→ (H1 , . . . , Hk )
Pf: If G → (H1 , . . . , Hk ), we delete edges as long as the deletion does
not cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is
(H1 , . . . , Hk )-Ramsey minimal
I
For any e ∈ E (G ), G + e → (H1 , . . . , Hk )
Rmin (H1 , . . . , Hk ) Saturation
G is Rmin (H1 , . . . , Hk ) saturated iff
G 6→ (H1 , . . . , Hk )
For any e ∈ E (G ), G + e → (H1 , . . . , Hk )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
5 / 11
Saturation of Rmin (Kk1 , . . . , Kkt )
Example
Let r := r (k1 , . . . , kt ) be the Ramsey number of (Kk1 , . . . , Kkt ). Then
Kr −2 ∨ Ks
is Rmin (Kk1 . . . , Kkt ) saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
6 / 11
Saturation of Rmin (Kk1 , . . . , Kkt )
Example
Let r := r (k1 , . . . , kt ) be the Ramsey number of (Kk1 , . . . , Kkt ). Then
Kr −2 ∨ Ks
is Rmin (Kk1 . . . , Kkt ) saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
6 / 11
Saturation of Rmin (Kk1 , . . . , Kkt )
Example
Let r := r (k1 , . . . , kt ) be the Ramsey number of (Kk1 , . . . , Kkt ). Then
Kr −2 ∨ Ks
is Rmin (Kk1 . . . , Kkt ) saturated.
Kr −2 ∨ Ks 6→ (Kk1 , . . . , Kkt )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
6 / 11
Saturation of Rmin (Kk1 , . . . , Kkt )
Example
Let r := r (k1 , . . . , kt ) be the Ramsey number of (Kk1 , . . . , Kkt ). Then
Kr −2 ∨ Ks
is Rmin (Kk1 . . . , Kkt ) saturated.
Kr −2 ∨ Ks 6→ (Kk1 , . . . , Kkt )
Ferrara-Kim-Yeager (UCD, UIUC)
Kr −2 ∨ Ks + e → (Kk1 , . . . , Kkt )
Saturation of Ramsey-Minimal Families
29 March 2014
6 / 11
Saturation of Rmin (Kk1 , . . . , Kkt )
Example
Let r := r (k1 , . . . , kt ) be the Ramsey number of (Kk1 , . . . , Kkt ). Then
Kr −2 ∨ Ks
is Rmin (Kk1 . . . , Kkt ) saturated.
Corollary
sat(n; Rmin (Kk1 , . . . , Kkt )) ≤
Ferrara-Kim-Yeager (UCD, UIUC)
r −2
2
+ (r − 2)(n − r + 2) when n ≥ r
Saturation of Ramsey-Minimal Families
29 March 2014
6 / 11
Saturation of Rmin (Kk1 , . . . , Kkt )
Example
Let r := r (k1 , . . . , kt ) be the Ramsey number of (Kk1 , . . . , Kkt ). Then
Kr −2 ∨ Ks
is Rmin (Kk1 . . . , Kkt ) saturated.
Corollary
sat(n; Rmin (Kk1 , . . . , Kkt )) ≤
r −2
2
+ (r − 2)(n − r + 2) when n ≥ r
Hanson-Toft Conjecture, 1987
sat(n; Rmin (Kk1 , . . . , Kkt )) =
Ferrara-Kim-Yeager (UCD, UIUC)
r −2
2
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Saturation of Ramsey-Minimal Families
29 March 2014
6 / 11
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Saturation of Ramsey-Minimal Families
29 March 2014
7 / 11
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n; Rmin (K3 , K3 )) =
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<6=r
4n − 10 n ≥ 56
Saturation of Ramsey-Minimal Families
29 March 2014
7 / 11
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n; Rmin (K3 , K3 )) =
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<6=r
4n − 10 n ≥ 56
Saturation of Ramsey-Minimal Families
29 March 2014
7 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
8 / 11
Sat Number of Ramsey-Minimal Families of Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
9 / 11
Sat Number of Ramsey-Minimal Families of Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
Corollary
sat(n; Rmin (k1 K2 + · · · + kt K2 )) ≤ 3(k1 + · · · + kt − t)
when n ≥ 3(k1 + · · · + kt − t)
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
9 / 11
Sat Number of Ramsey-Minimal Families of Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
Corollary
sat(n; Rmin (k1 K2 + · · · + kt K2 )) ≤ 3(k1 + · · · + kt − t)
when n ≥ 3(k1 + · · · + kt − t)
Ferrara, Kim, Y.; 2014
sat(n; Rmin (k1 K2 + · · · + kt K2 )) = 3(k1 + · · · + kt − t)
when n > 3(k1 + · · · + kt − t)
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
9 / 11
Sat Number of Ramsey-Minimal Families of Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
Corollary
sat(n; Rmin (k1 K2 + · · · + kt K2 )) ≤ 3(k1 + · · · + kt − t)
when n ≥ 3(k1 + · · · + kt − t)
Ferrara, Kim, Y.; 2014
sat(n; Rmin (k1 K2 + · · · + kt K2 )) = 3(k1 + · · · + kt − t)
when n > 3(k1 + · · · + kt − t)
Construction is generally unique: vertex-disjoint triangles with isolates.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
9 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
10 / 11
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Corollary
If G is Rmin (H1 , . . . , Hk ) saturated, then G = G1 ∪ · · · ∪ Gk , where Gi is
Hi saturated and all Gi share the same vertex set.
Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
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Thanks for Listening!
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Ferrara-Kim-Yeager (UCD, UIUC)
Saturation of Ramsey-Minimal Families
29 March 2014
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