Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory

Definition 3.1

Let R be a commutative ring, n ∈ ℕ ≥ 2 and S = { m 1 , … , m n } ⊆ R • \ U ( R ) be a set of n pairwise coprime non-zero non-units and m = m 1 m 2 ⋯ m n . (Note that m = 0 is possible.)

1. We say d is a perfect divisor of m with respect to S if d ≠ m and d is a product of distinct elements of S.

2. The perfect divisor graph P D G ( S ) of S is defined as the simple, undirected graph ( V , E ) , where V = { d | d perfect-divisor of m } is the vertex set and for two vertices a ≠ b ∈ V , ( a , b ) ∈ E if and only if a | b or b | a .

3. By P D G we denote the class of all perfect divisor graphs.

Lemma 3.2

Let R be a commutative ring, n ∈ ℕ ≥ 2 and S = { m 1 , … , m n } ⊆ R • \ U ( R ) be a set of n pairwise coprime non-zero non-units and m = m 1 m 2 ⋯ m n . Furthermore, let

and define ≤ on V such that for all a, b ∈ V , we have a ≤ b if and only if a = b or a | b .

Then ( V , ≤ ) is a partially ordered set of cardinality | V | = 2 n − 2 and PDG ( S ) is a partial order graph.

Proof

The relation ≤ clearly is reflexive and transitive, we prove that it is also antisymmetric. Let d ∈ V be a perfect divisor of m with respect to S. Then d = ∏ j ∈ J m j for ∅ ≠ J ⊆ { 1 , … , n } . We show that for every 1 ≤ i ≤ m , m i | d if and only if i ∈ J .

Obviously, if j ∈ J , then m i | d . Let us assume that i ∈ { 1 , … , n } \ J . Then by hypothesis, for j ∈ J there are elements a j and b j ∈ R such that a j m j + b j m i = 1 holds. Hence,

for some a, c ∈ R . Therefore, d and m i are coprime elements of R which in particular implies that m i ∤ d .

It follows that if d 1 and d 2 are distinct perfect divisors of m and d 1 | d 2 , then d 2 ∤ d 1 . Thus, ( V , ≤ ) is a partially ordered set.

Moreover, it follows that the elements in V correspond to the non-empty proper subset of { 1 , … , n } . Therefore, their number amounts to

Theorem 3.3

Let R be a commutative ring, n ∈ ℕ ≥ 2 and S = { m 1 , … , m n } ⊆ R • \ U ( R ) be a set of n pairwise coprime non-zero non-units, m = m 1 m 2 ⋯ m n and P D G ( S ) the perfect divisor graph of m with respect to S.

Then the following assertions hold:

1. PDG ( S ) is a connected graph if and only if n ≥ 3 .

2. If n ≥ 3 , then the diameter diam ( PDG ( S ) ) = 3 .

3. The domination number of PDG ( S ) is equal to 2 if n ≥ 2 and equal to 1 if n = 1 .

4. If n ≥ 3 , then the vertices in P k = { ∏ j ∈ J m j | | J | = k } for 1 ≤ k ≤ n − 1 are pairwise not connected by an edge. In particular, PDG ( S ) is an ( n − 1 ) -partite graph.

5. If a ∈ P k = { ∏ j ∈ J m j | | J | = k } for 1 ≤ k ≤ n − 1 , then deg ( a ) = 2 k + 2 n − k − 4 .

6. If n ≥ 3 , then for the girth of PDG ( S ) the following holds

   g ( PDG ( S ) ) = 6 n = 3 , 3 n ≥ 4 .

7. PDG ( S ) is planar if and only if n ∈ { 3 , 4 } .

Proof

(1) If n = 2 , then V consists of two vertices m 1 and m 2 which are coprime and hence not connected. Assume n ≥ 3 and let a = ∏ j ∈ J m j and b = ∏ k ∈ K m k be two distinct vertices of PDG ( S ) . Suppose that m j = m k for some j ∈ J and k ∈ K . Then a − m j − b is a path of length 2 from a to b if m j ≠ a , b and ( a , b ) is an edge otherwise. Suppose that m j ≠ m k for every j ∈ J and k ∈ K . We show that | J | ≤ n − 2 or | K | ≤ n − 2 . Suppose that | J | = | K | = n − 1 . Since n ≥ 3 and m j ≠ m k for every j ∈ J and k ∈ K , we conclude that | { m j | j ∈ J } ∪ { m k | k ∈ K } | = 2 n − 2 > n , a contradiction. Thus, | J | ≤ n − 2 or | K | ≤ n − 2 . Without loss of generality, we may assume that | J | ≤ n − 2 . Take arbitrary k ∈ K . Then, a − a m k − m k − b is a path of length 3 from a to b if b ≠ m k and otherwise, a − a m k − b is a path of length 2. Hence, PDG ( S ) is connected which completes the proof of (1).

(2) Suppose that n ≥ 3 . Then PDG ( S ) is connected by (1). Let a , b be two distinct vertices of PDG ( S ) . In light of the proof given in (1), we have d ( a , b ) ≤ 3 . Let a = ∏ j = 1 ( n − 1 ) m j and b = m n . Then a − m 1 − m 1 b − b is a shortest path in PDG ( S ) from a to b. Hence, d ( a , b ) = 3 . Thus, diam ( PDG ( S ) ) = 3 .

For (3) observe that every perfect divisor d of m is either divisible by m 1 or divides m 2 m 3 ⋯ m n . Hence, every vertex of PDG ( S ) is connected by an edge to either one of these two vertices.

(4) Let 1 ≤ k ≤ n − 1 and J, K ⊆ { 1 , … , n } with | J | = | K | = k . Set a = ∏ j ∈ J m j and b = ∏ k ∈ K m k be two different vertices of PDG ( S ) , which implies J ≠ K . Therefore, there exist j ∈ J \ K and k ∈ K \ J . In the proof of Lemma 3.2, we have shown that it now follows that m j ∤ b and m k ∤ a . In particular, it follows that a ∤ b and b ∤ a . Hence, no two vertices in { ∏ j ∈ J m j | | J | = k } are connected by an edge.

For (5), let a = ∏ j ∈ J m j be perfect divisor of m and set k = | J | . The perfect divisors of m with respect to S which divide a which are connected by an edge to a correspond to the non-empty, proper subsets of J which are ∑ i = 1 k − 1 k i = 2 k − 2 many. In addition, we need to count the number of perfect divisors of m which are divisible by a. These are exactly the ones of the form ∏ k ∈ K m k with J ⊊ K ⊊ { 1 , … , n } of which there are ∑ i = 1 n − k − 1 n − k i = 2 n − k − 2 . Hence, deg ( a ) = 2 k + 2 n − k − 4 .

(6) For n = 3 , we can verify in Figure 1 that there is cycle of length 6 and no shorter cycle.

If n ≥ 4 , then m 1 m 2 m 3 is a perfect divisor and the edges ( m 1 , m 1 m 2 ) , ( m 1 m 2 , m 1 m 2 m 3 ) and ( m 1 m 2 m 3 , m 1 ) form a cycle of length 3 which is the smallest possible length of a cycle in PDG ( S ) .

Finally, for (7), it is easily verified that PDG ( S ) is planar if n = 3 , cf. Figure 1. Moreover, Figure 2 shows a planar arrangement of the edges of PDG ( S ) for n = 4 .

If, however, n ≥ 5 , then Figure 3 is a K 3 , 3 subgraph of PDG ( S ) , and hence PDG ( S ) is not a planar by Kuratowski’s Theorem on planar graphs.□

Theorem 3.4

Let n, m ≥ 1 . Then for the Ramsey number ℛ P D G with respect to the class P D G of perfect divisor graphs the following holds:

Proof

We set w = ( n − 1 ) ( m − 1 ) and show that ℛ P D G ( n , m ) > w = ( n − 1 ) ( m − 1 ) by giving an example of perfect divisor graph G and an induced subgraph H of G with w vertices which is a complete ( n − 1 ) -partite graph on w vertices in which independent sets are of cardinality at most m − 1 .

Let R = Z and let S = { p 1 , p 2 , … , p w } be a set of w distinct positive prime numbers of Z . We set m = p 1 p 2 ⋯ p w and G = PDG ( S ) .

For each 1 ≤ i ≤ n − 1 , let k i = ( i − 1 ) ( m − 1 ) and we set a i = p 1 p 2 ⋯ p k i (where a 1 = 1 ) and

Note that A 1 = { p 1 , … , p m − 1 } .

Let H be the subgraph of G induced by the vertex set A 1 ∪ A 2 ∪ ⋯ ∪ A n − 1 . By construction, for each 1 ≤ i ≤ n − 1 , | A i | = m − 1 holds and A i is contained in the partition P k i + 1 of G, cf. Theorem 3.3.4. This implies that each A i is an independent vertex set of H of cardinality m − 1 .

Moreover, since G is a ( w − 1 ) -partite graph and each A i is contained in P k i + 1 , it follows that H is an ( n − 1 ) -partite graph (with partitioning A 1 ∪ A 2 ∪ ⋯ ∪ A n − 1 ). For an example of this construction with m = 5 and n = 4 see Example 3.5.

Thus, not more than m − 1 vertices of H are independent, and a straight-forward verification shows that the clique number of H is at most n − 1 . Thus, ℛ P D G ( n , m ) > w . Hence, by Theorem 2.2, we have ℛ P D G ( n , m ) = ℛ P o G ( n , m ) = w + 1 = ( n − 1 ) ( m − 1 ) + 1 .□

Example 3.5

We demonstrate the construction of the previous proof for the example R = Z with n = 4 and m = 5 . That is, we construct a perfect divisor graph which has a complete 3-partite graph H as subgraph and each of the partitions of H consist of four independent vertices.

Definition 3.6

Let R be a commutative ring and a, b be distinct elements of R.

1. If a is a non-zero non-unit element of R, then we say a is a proper element of R.

2. If a | b (in R) and b ∤ a (in R), then we write a | | b .

3. The divisibility graph Div ( R ) of R is the undirected simple graph whose vertex set consists of the proper elements of R such that two vertices a ≠ b are adjacent if and only if a | | b or b | | a .

Lemma 3.7

Let R be a commutative ring and let V be the set of all proper elements of R and define ≤ on V such that for all a, b ∈ V , we have a ≤ b if and only if a = b or a | | b .

Then ( V , ≤ ) is a partially ordered set and the divisibility graph Div ( R ) of R is a partial order graph.

Theorem 3.8

Let n, m ≥ 1 be positive integers (n, m need not be distinct). Then for the Ramsey number ℛ D i v G with respect to the class D i v G of divisibility graphs the following holds:

Corollary 3.9

Let n, m ≥ 1 be positive integers (n, m need not be distinct), k = ( n − 1 ) ( m − 1 ) + 1 , R be a commutative ring and S be a subset of proper elements of R such that | S | ≥ k .

Then one of the following assertions holds:

1. There are n elements a 1 , … , a n ∈ S such that a 1 | | a 2 | | ⋯ | | a n (in R).

2. There are m pairwise distinct elements b 1 , … , b m ∈ S such that for all 1 ≤ h ≠ f ≤ m either

   • b h ∤ b f or

   • b h | b f and b f | b h

hold.

Definition 3.10

Let R be a ring.

1. We call a left (right) ideal I of R non-trivial if I ≠ { 0 } and I ≠ R .

2. The inclusion ideal graph In ( R ) of R is the (simple, undirected) graph whose vertex set is the set of non-trivial left ideals of R and two distinct left ideals I, J are adjacent if and only if I ⊂ J or J ⊂ I (cf. Akbari et al. [10]).

3. By I n G , we denote the class of all inclusion ideal graphs.

Remark 3.11

The set V of all non-trivial left ideals of a ring R together with the partial order ⊆ induced by inclusion is a partially ordered set. Hence, the inclusion graph In ( R ) of a ring R is a partial order graph.

Theorem 3.12

Let n, m ≥ 1 be positive integers (n, m need not be distinct). Then for the Ramsey number ℛ I n G with respect to the class I n G of inclusion ideal graphs the following holds:

Corollary 3.13

Let R be a ring, n, m ≥ 1 be positive integers (n, m need not be distinct) and S ⊆ { I | I i s a n o n - t r i v i a l l e f t i d e a l o f R } such that | S | ≥ ( n − 1 ) ( m − 1 ) + 1 .

Then one of the following assertions hold:

1. There are n pairwise distinct elements (non-trivial left ideals) I 1 , … , I n ∈ S with I 1 ⊂ I 2 ⊂ ⋯ ⊂ I n .

2. There are m elements (non-trivial left ideals) J 1 , … , J m ∈ S such that J a ⊈ J b for every 1 ≤ a ≠ b ≤ m .

Definition 3.14

Let R be a commutative ring which is not a field and j ≥ 2 an integer.

1. We denote by R j × j the ring of all j × j matrices with entries in R.

2. Let V = { A ∈ R j × j | det ( A ) a proper element of R } be the set of all j × j matrices whose determinant is a proper element of R, cf. Definition 3.6. We define the matrix graph Mat G ( R ) of R to be the undirected simple graph with V as its vertex set and two distinct vertices A, B ∈ V are adjacent if and only if det ( A ) | | det ( B ) or det ( A ) | | det ( B ) .

3. By ℳ a t G we denote the class of all matrix graphs.

Lemma 3.15

Let R be a commutative ring which is not a field, j ≥ 2 an integer and

Define ≤ on V such that for all A, B ∈ V , we have A ≤ B if and only if A = B or det ( A ) | | det ( B ) .

Then ( V , ≤ ) is a partially ordered set and the graph MatG ( R ) is a partial order graph.

Theorem 3.16

Let n, m ≥ 1 be positive integers (n, m need not be distinct). Then for the Ramsey number ℛ ℳ a t G with respect to the class ℳ a t G of matrix graphs the following holds:

Proof

Let R = Z and j ≥ 2 and set w = ( n − 1 ) ( m − 1 ) ≥ 1 . Furthermore, let p 1 , p 2 , … , p w be distinct positive prime numbers of Z and choose X i ∈ R j × j with det ( X i ) = p i for 1 ≤ i ≤ w .

We construct a matrix graph MatG ( R ) which has a complete ( n − 1 ) -partite subgraph H in which each partition has m − 1 vertices. The construction is analogous to the one in the proof of Theorem 3.4.

For each 1 ≤ i ≤ n − 1 , let k i = ( i − 1 ) ( m − 1 ) , q i = X 1 X 2 ⋯ X k i (hence q 1 = I j the identity matrix j × j ) and

Note that A 1 = { X 1 , … , X m − 1 } . Since det ( q i X k i + j ) = p 1 … p k i p k i + j , it follows that the elements of A i are pairwise distinct and | A i | = m − 1 for 1 ≤ i ≤ n − 1 .

Let S = A 1 ∪ A 2 ∪ ⋯ ∪ A n − 1 and set G = MatG ( Z ) . Then for each i, the vertices in A i are independent. However, there are edges between all vertices of two distinct sets A i and A j with i ≠ j . Therefore, G is a complete ( n − 1) -partite graph in which each partition has m − 1 vertices that are independent. Thus, at most m − 1 vertices of G are independent. It is easily verified that the clique number of G is n − 1 . It follows that ℛ ℳ a t G ( n , m ) > w and together with Theorem 2.2 we conclude ℛ ℳ a t G ( n , m ) = ℛ P o G ( n , m ) = w + 1 = ( n − 1 ) ( m − 1 ) + 1 .□

Corollary 3.17

Let R be a commutative ring, j ≥ 2 , n, m ≥ 1 be positive integers (n, m need not be distinct) and S ⊆ { X ∈ D | det ( X ) b e a p r o p e r e l e m e n t o f R } such that | S | ≥ ( n − 1 ) ( m − 1 ) + 1 .

Then one of the following assertions hold:

1. There are n matrices X 1 , … , X n ∈ S such that det ( X 1 ) | | det ( X 2 ) | | ⋯ | | det ( X n ) (in R).

2. There are m pairwise distinct matrices Y 1 , … , Y m ∈ S such that for all 1 ≤ h ≠ f ≤ m .

   • det ( Y h ) ∤ det ( Y f ) or

   • det ( Y h ) | det ( Y f ) and det ( Y f ) | det ( Y h )

hold.

Definition 3.18

Let R be a commutative ring.

1. We call a ∈ R idempotent if a 2 = a .

2. We define the idempotent graph Idm ( R ) of R to be the undirected simple graph with the set of idempotents of R as its vertex set and two distinct vertices a, b are adjacent if and only if a | b or b | a .

3. By I d e m G we denote the class of all idempotent graphs.

Lemma 3.19

Let R be a commutative ring and let V be the set of all idempotent elements of R. We define ≤ on V such that for all a, b ∈ V , we have a ≤ b if and only if a | b .

Then ( V , ≤ ) is a partially ordered set and the graph Idm ( R ) is a partial order graph.

Proof

Clearly, ≤ is reflexive and transitive. Suppose that a | b and b | a (in R), that is, a = b x and b = a y for some x, y ∈ R . Then, since a and b are idempotent, we can conclude that

and hence a = b a = a b = b , which implies that ≤ is anti-symmetric.□

Lemma 3.20

Let R be a commutative ring and E be a set of w ≥ 3 distinct non-trivial idempotents of R such that eR is a maximal ideal of R for every e ∈ E . Let x = f 1 f 2 ⋯ f k and y = b 1 b 2 ⋯ b j such that f 1 , … , f k , b 1 , … , b j ∈ E and 2 ≤ k , j < w .

Then

1. x ≠ 0 .

2. x = y if and only if { f 1 , … , f k } = { b 1 , … , b j } .

Proof

1. Since e 1 , … , e w are distinct non-trivial idempotents of R and each e i R is a maximal ideal of R, 1 ≤ i ≤ w , by Lemma 3.19 we conclude that e 1 R , … , e w R are distinct maximal ideals of R. Since k < w , there exists a maximal ideal dR for some d ∈ E such that x = f 1 f 2 ⋯ f k ∉ d R (note that each f i R is a maximal ideal of R). Thus, x ≠ 0 .

2. We may assume that f 1 ≠ b i for every 1 ≤ i ≤ j . Hence, x ∈ f 1 R but y ∉ f 1 R and thus x ≠ y . Since all f i and b i are idempotent elements, multiplicities have no impact, which makes the other implication obvious.□

Theorem 3.21

Let n, m ≥ 1 be positive integers (n, m need not be distinct). Then for the Ramsey number ℛ I d e m G ( n , m ) with respect to the class of idempotent graphs the following holds:

Proof

We set w = ( n − 1 ) ( m − 1 ) ≥ 1 and show that I d e m G contains an ( n − 1 ) -partite graph in which each partition consists of m − 1 independent vertices. For this purpose, set R = ∏ i = 1 w Z 2 . It is clear that R has exactly w distinct maximal ideals, say M 1 , … , M w , and each M i = p i R , 1 ≤ i ≤ w for idempotent p i of R. We set E = { p 1 , p 2 , … , p w } . Note that | E | = w since p 1 , p 2 , … , p w are pairwise distinct.

For each 1 ≤ i ≤ n − 1 , let n i = ( i − 1 ) ( m − 1 ) , a i = p 1 p 2 ⋯ p n i (hence a 1 = 1 ) and A i = { a i p n i + 1 , … , a i p n i + ( m − 1 ) } . Note that A 1 = { p 1 , … , p m − 1 } .