Some results on the Ryser design conjecture-III
aa r X i v : . [ m a t h . C O ] N ov SOME RESULTS ON THE RYSER DESIGN CONJECTURE-III
TUSHAR D. PARULEKAR AND SHARAD S. SANE abstract.
A Ryser design D on v points is a collection of v proper subsets (called blocks)of a point-set with v points such that every two blocks intersect each other in λ points (and λ < v is a fixed number) and there are at least two block sizes. A design D is called asymmetric design, if every point of D has the same replication number (or equivalently, allthe blocks have the same size) and every two blocks intersect each other in λ points. Theonly known construction of a Ryser design is via block complementation of a symmetricdesign. Such a Ryser design is called a Ryser design of Type-1. This is the ground for theRyser-Woodall conjecture: “every Ryser design is of Type-1”. This long standing conjecturehas been shown to be valid in many situations. Let D denote a Ryser design of order v ,index λ and replication numbers r , r . Let e i denote the number of points of D withreplication number r i (with i = 1 , A of D small (respectively large) if | A | < λ (respectively | A | > λ ) and average if | A | = 2 λ . Let D denote the integer e − r and let ρ > r − r − Introduction
A design is a pair (
X, L ), where X is a finite set of points and L ⊆ P ( X ), where P ( X ) isthe power set of X . The elements of X are called its points and the members of L are calledthe blocks. Most of the definitions, formulas and proofs of standard results used here can befound in [3]. Definition 1.1.
A design D = ( X, L ) is said to be a symmetric ( v, k, λ ) design if1. | X | = | L | = v ,2. | B ∩ B | = λ ≥ B and B of D , B = B ,3. | B | = k > λ for all blocks B of D . Date : November 18, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Ryser design.
Definition 1.2.
A design D = ( X, L ) is said to be a Ryser design of order v and index λ ifit satisfies the following:1. | X | = | L | = v ,2. | B ∩ B | = λ for all blocks B and B of D , B = B ,3. | B | > λ for all blocks B of D ,4. there exist blocks B and B of D with | B | 6 = | B | .Here condition 4 distinguishes a Ryser design from a symmetric design, and condition 3disallows repeated blocks and also any block being contained in another block.Woodall [11] introduced a type of combinatorial object which is a combinatorial dual of aRyser design. All the known examples of Ryser designs can be described by the followingconstruction which is also known as the Ryser-Woodall complementation.Let D = ( X, A ) be a symmetric ( v, k, k − λ ) design with k = 2 λ . Let A be a fixed block of D . Form the collection B = { A } S { A △ B : B ∈ A , B = A } , where A △ B denotes the usualsymmetric difference of A and B . Then D = ( X, B ) is a Ryser design of order v and index λ obtained from D by block complementation with respect to the block A . We denote D by D ∗ A . Then A is also a block of D ∗ A and the original design D can be obtained bycomplementing D ∗ A with respect to the block A . If D is a symmetric ( v, k, λ ′ ) design, thenthe design obtained by complementing D with respect to some block is a Ryser design oforder v with index λ = k − λ ′ . A Ryser design obtained in this way is said to be of Type-1 .Define a Ryser design to be of
Type-2 if it is not of Type-1. We now state
The Ryser Design Conjecture [3] : Every Ryser design is of Type-1.
In a significant paper Singhi and Shrikhande [9] proved the conjecture when the index λ isa prime. In [8] Seress showed the truthfulness of the conjecture for λ = 2 p , where p is aprime. In [2] Ionin and Shrikhande developed a new approach to the Ryser design conjecturethat led to new results for certain parameter values. They also gave an alternative proof ofthe celebrated non-uniform Fisher Inequality. Ionin and Shrikhande went on to explore thevalidity of the Ryser design conjecture from a different perspective. Their results prove theconjecture for certain values of v rather than for λ . Both Ryser and Woodall independentlyproved the following result: Theorem 1.3 ([3, Theorem 14.1.2] Ryser Woodall Theorem) . If D is a Ryser design oforder v , then there exist integers r and r , r = r such that r + r = v + 1 and any pointoccurs either in r blocks or in r blocks. The point-set is partitioned into subsets E and E , where E i is the set of points withreplication number r i and let e i = | E i | for i = 1 ,
2. Then e , e > e + e = v . For ablock A , let τ i ( A ) denote | E i ∩ A | , the number of points of block A with replication number r i for i = 1 ,
2. Then | A | = τ ( A ) + τ ( A ). We say a block A is large, average or small if | A | is greater than λ , equal to λ or less than λ respectively. A block which is not average is OME RESULTS ON THE RYSER DESIGN CONJECTURE-III 3 called a non average block.
The Ryser-Woodall complementation (or block complementation) of a Ryser design D ofindex λ with respect to some block A ∈ D is either a symmetric design or a Ryser designof index ( | A | − λ ). If D ∗ A is the new Ryser design of index ( | A | − λ ) obtained by Ryser-Woodall complementation of a Ryser design D with respect to the block A , we denote thenew parameters of D ∗ A by λ ( D ∗ A ) , e ( D ∗ A ) etc.Let D r ( X ) denote the set of all incidence structures D = ( X, B ) where B is a set of subsets of X and D is a Ryser design with replication numbers r and r = v + 1 − r ; or a symmetricdesign with block size r or r . Proposition 1.4 ([3, Proposition 14.1.7]) . Let
D ∈ D r ( X ) and let A, B be blocks of D .Then D ∗ A ∈ D r ( X ) and the following conditions hold: (i) ( D ∗ A ) ∗ A = D ; (ii) A △ B is a block of D ∗ A and ( D ∗ A ) ∗ ( A △ B ) = D ∗ ( B ) ; (iii) r ( D ∗ A ) = r ( D ) ; (iv) λ ( D ∗ A ) = | A | − λ ( D ) ; (v) E ( D ∗ A ) = E ( D ) △ A ; (vi) e ( D ∗ A ) = e ( D ) − τ ( A )( D ) + τ ( A )( D ) ; (vii) D ∗ A is a symmetric design if and only if A = E ( D ) or A = E ( D ) . Remark 1.5.
Since | A △ B | = | A | + | B | − | A ∩ B | , observe that if a design is of Type-1then it has all average blocks except for one, and hence a Type-2 Ryser design must have atleast two non average blocks. Following Singhi and Shrikhande [9] we define ρ = r − r − cd , where gcd( c, d ) = 1 . Let g = gcd( r − , r − . Then r + r = v + 1 implies g divides ( v − r − cg, r − dg and v − c + d ) g . We also write a to denote c − d and observe that any two of c, d and a are coprime. We use the following equations which can be found in [9] and [2].In a Ryser design with block sizes k , k , .....k v (1) v X m =1 k m − λ = ( ρ + 1) ρ − λe r ( r −
1) + e r ( r −
1) = λv ( v − ρ − e = λ ( ρ + 1) − r (3) ( ρ − e = ρr − λ ( ρ + 1)(4)Let A be a block of a Ryser design D with | A | = τ ( A ) + τ ( A ), a simple two way countinggives, ( r − τ ( A ) + ( r − τ ( A ) = λ ( v − TUSHAR D. PARULEKAR AND SHARAD S. SANE which gives τ ( A ) = λ − tdτ ( A ) = λ + tc | A | = 2 λ + ta for some integer t . These findings are summed in the following lemma. Lemma 1.6.
Let A be a block of a Ryser design. Then the size of A has the form | A | =2 λ + ta , where t is an integer. The block A is large, average or small depending on whether t > , t = 0 or t < respectively. Hence τ ( A ) = τ ( A ) = λ if A is an average block, τ ( A ) > λ > τ ( A ) if A is a small block and τ ( A ) > λ > τ ( A ) if A is a large block. In a Ryser design D with blocks sizes | A i | = k i for i = 1 , , . . . , v the column sum of theincidence matrix is equal to the row sum of the incidence matrix which implies P k i = e r + e r . Hence from equation (3) and (4), we get(5) e r + e r = λ ( v −
1) + r r In this article, a binary set operation △ is defined that gives an equivalence relation on theset of Ryser designs of order v . Some observations on the block complementation procedureof Ryser-Woodall are made. It is shown that a Ryser design of order v and index λ withtwo block sizes and with one block size 2 λ is of Type-1. It is also shown that, under theassumption that large and small blocks do not coexist in any Ryser design equivalent to agiven Ryser design, the given Ryser design must be of Type-1.2. Equivalence classes of Ryser designs
The binary set operation A △ B = ( A ∩ B c ) ∪ ( A c ∩ B ) is well known and A △ B is the setof all the elements that are in precisely one of the sets A and B . The following (Booleanalgebraic) lemma is also well known and hence the elementary proof is omitted. Lemma 2.1.
The binary set operation A △ B has the following properties:(a) △ is commutative. Further, A △ B = A if and only if B = ∅ . Also A △ A = ∅ .(b) △ is associative. In fact the set A △ A △· · ·△ A n precisely consists of those elementsthat belong to an odd number of A i s. Let V be a v -set ( which is now fixed for the entire discussion to follow ). Let Ω be a familyof v distinct non-empty subsets of V . Members of Ω are called its blocks . Let A ∈ Ω. Definea function (block complementation w.r.t. A ) f A on Ω as follows: f A ( A ) = A and for all B = A define f A ( B ) = A △ B . Then Ω ′ = f A (Ω) is also a family of v subsets of V . We alsoemphasize that f A is not defined on Ω if A / ∈ Ω. The following lemma is then obvious.
Lemma 2.2.
With everything as above, let Ω ′ = f A (Ω) . Then: OME RESULTS ON THE RYSER DESIGN CONJECTURE-III 5 (a) A ∈ Ω ′ .(b) Ω ′ is also a collection of v distinct non-empty subsets of Ω .(c) f A (Ω ′ ) = Ω . Lemma 2.3.
Let f A (Ω) = Ω ′ and let f A △ B (Ω ′ ) = Ω ′′ where A and B are distinct subsets in Ω . Then Ω ′′ = f B (Ω) .Proof. Let Ω ∗ = f B (Ω). Then besides B , Ω ∗ contains all the sets of the form B △ C where C ∈ Ω and C = B . Let C ∈ Ω and C = A, B . Then f A △ B f A ( C ) = f A △ B ( A △ C ) = ( A △ B ) △ ( A △ C ) = ( B △ C )Further, f A △ B f A ( A ) = f A △ B ( A ) = ( A △ B ) △ A = B and f A △ B f A ( B ) = f A △ B ( A △ B ) = A △ B . This shows that Ω ∗ = Ω ′′ . (cid:3) We also define a generic universal function on a family Ω of subsets of V : g ( A ) = A forevery A ∈ Ω. Evidently, g (Ω) = Ω and g is valid (properly defined) on any Ω. Theorem 2.4.
Let Ω be a family of v distinct non-empty subsets of V .(a) Consider the following diagram: Ω = Ω h −→ Ω h −→ Ω h −→ · · · · · · h n −→ Ω n = Ω ′ where each h i equals g or f A i and the functions are valid on the families they aredefined. Then Ω ′ = g (Ω) = Ω or Ω ′ = f A (Ω) for some A ∈ Ω .(b) Let X be the set of all Ω ′ that can be obtained from Ω by a sequence of functions as in(a) and let Y be the set of all Ω ′′ that can be obtained by a single function h (Ω) = Ω ′′ where h = g or h = f A for some A ∈ Ω . Then X = Y .(c) For two families Ω and Σ of v distinct nonempty subsets of V , write Σ ∼ Ω if Σ = h (Ω) where h = g or h = f A for some A ∈ Ω . Then ∼ is an equivalent relation.Proof. If some h i = g then we can effectively reduce the length of the sequence. Also, ifΩ ′ = Ω n = Ω, then Ω ′ = g (Ω) and we are done. The proof of (a) is by induction on n . Thestatement clearly holds for n = 1. LetΩ = Ω h −→ Ω h −→ Ω h −→ · · · · · · h n −→ Ω n = Ω ′ h n +1 −−−→ Ω n +1 = Ω ′′ If Ω ′ = Ω, then Ω ′′ = g (Ω) or Ω ′′ = f A (Ω) for some A ∈ Ω and we are done. Let Ω ′ = Ω.Then by the induction hypothesis, Ω ′ = f A (Ω) for some A ∈ Ω. If h n +1 = g , then Ω ′′ =Ω ′ = f A (Ω) and we are done. Otherwise, h n +1 = f A △ B (Ω ′ ) for some A △ B ∈ Ω ′ , thatis, Ω ′′ = f A △ B f A (Ω) = f B (Ω) for some B ∈ Ω (by Lemma 2.3) proving (a). Consider (b).Clearly Y ⊂ X . Using (a), if Ω ′ ∈ X , then either Ω ′ = Ω or Ω ′ = f A (Ω) for some A ∈ Ωshowing that Ω ′ ∈ Y . Hence, X = Y . Note that reflexivity and symmetry of ∼ are takencare of by the function g and the fact that f A ( f A (Ω)) = Ω and (a) clearly proves transitivity.Thus ∼ is indeed an equivalence relation proving (c). (cid:3) TUSHAR D. PARULEKAR AND SHARAD S. SANE
Corollary 2.5.
Let
Ω = { A , A , · · · , A v } be a family of v distinct nonempty subsets of V .Let Ω i = f A i (Ω) and let (Ω) denote the equivalence class of Ω . Then: (Ω) = { Ω } ∪ { Ω i : i = 1 , , · · · , v } Definition 2.6.
Let
Ω = { A , A , · · · , A v } be a family of v distinct nonempty subsets of V .For a point x ∈ V , let r ( x ) denote the replication number (the number of blocks containing x ) of x . Suppose we have constants r > r such that(i) r + r = v + 1 .(ii) r ( x ) = r or r ( x ) = r for every x ∈ V (the possibility of all replication numbersbeing equal is also admissible).Then Ω is called a Ryser system with parameter triple ( v, r , r ) . Theorem 2.7.
Let Ω be a Ryser system with parameter triple ( v, r , r ) . Then for every Ω ′ ∈ (Ω) the Ryser system Ω ′ also has the same parameter triple.Proof. Let f A (Ω) = Ω ′ . If x / ∈ A , then r Ω ′ ( x ) = r Ω ( x ) which equals r or r since Ω is aRyser system. Let x ∈ A and suppose w.l.o.g. that r Ω ( x ) = r . Then f A (Ω) has exactly( v − − ( r −
1) = r − A itself) that contain x . Hence r Ω ′ ( x ) = ( r −
1) + 1 = r . (cid:3) Theorem 2.8 ([3, Proposition 14.1.7]) . Let D be a Ryser design on v points. Then it isalso a Ryser system with parameter triple ( v, r , r ) and all Ryser designs in the equivalenceclass of D have the same parameter triple ( v, r , r ) . Theorem 2.9.
Let D be a Ryser design on v points. Then the following conditions areequivalent:(a) The equivalence class of D (under ∼ ) contains a symmetric design E .(b) D is of Type-1.(c) Every Ryser design in the equivalence class of D is of Type-1.Proof. Let D ∗ be a symmetric design in the equivalence class of D and let D ′ be some Ryserdesign in the equivalence class of D . Then there is some block A in D ∗ such that blockcomplementation with respect to that block produces D ′ . This proves that (a) implies both(b) and (c). Clearly (c) is stronger than (b). Finally equivalence of the relation ∼ showsthat (b) implies (a). (cid:3) Theorem 2.10.
Assume the following hypothesis: Every Ryser design D that has a block C of even size is of Type-1. Then the Ryser design conjecture is true.Proof. Since the Ryser design conjecture holds for small values of λ , we may assume that thegiven Ryser design D is one with λ ≥ λ ≥
2. Since every block must have size ≥ λ it follows(pigeonhole principle) that we have two blocks A and B of the same size k . Complementing OME RESULTS ON THE RYSER DESIGN CONJECTURE-III 7 D w.r.t. A then renders A △ B to have size 2( k − λ ) which is even. Since this is a block inan equivalent design E , the hypothesis implies E is of Type-1 and therefore by Theorem 2.9 D is also of Type-1. (cid:3) The main resultsTheorem 3.1.
Let D be a Ryser design of Type-2 of order v and index λ . Let A be a nonaverage block. Let D = D ∗ A be the new Ryser design of order v and index λ = ( | A | − λ ) obtained from D by block complementation with respect to the block A . If A is a large(respectively small) block in D then A is a small (respectively large) block in D . Let B beany other block of D and let B = A △ B be the new block in D obtained from B . Then:(i) If | B | > | A | (respectively | B | < | A | ) then B is a large (respectively small) block in D .(ii) If | B | = 2 λ then | B | = | A | in D .(iii) If | B | = | A | then | B | = 2 λ in D .Proof. Let | A | = k = 2 λ + ta for some integer t . Then A is a large block if t > A is asmall block if t <
0. If k > λ then λ > λ and 2 λ = 2( k − λ ) = k + ( k − λ ) > k . Hence if A is a large block of D then it becomes a small block in D . The other case is similar. Let B beany other block of D of size 2 λ + t ′ a . If t ′ = 0 then, B is an average block and if t ′ = 0 then, B is a non average block. In the new design D we have | B | = | A △ B | = | A | + | B | − | A ∩ B | .That is | B | = | A | + | B | − λ = 2( k − λ ) + | B | − | A | = 2 λ + | B | − | A | . This completes theproof. (cid:3) Lemma 3.2.
Let D be a Ryser design of order v and index λ . If D has v − blocks of size λ then D is of Type-1.Proof. Let A be the only non-average block of D of size k . Consider D = D ∗ A . Then byLemma 1.6 D has all v blocks of size k with block intersection k − λ . Hence D is a symmetric( v, k, λ ′ ) design with λ ′ = k − λ . Therefore D is a Ryser design of Type-1. (cid:3) Theorem 3.3.
Let D be a Ryser design of order v and index λ with two block sizes suchthat one block size is λ . Then D is of Type-1.Proof. Let the Ryser design D have two block sizes k and 2 λ where k = 2 λ . Let there be α blocks of size k and β blocks of size 2 λ , where α, β ≥ . Then we have(6) α + β = v Using (1) we get αk − λ + βλ = ( ρ + 1) ρ − λ . Hence we have(7) αk − λ + β + 1 λ = ( ρ + 1) ρ = ( v − ( r − r −
1) = ( v − r r − v TUSHAR D. PARULEKAR AND SHARAD S. SANE
Since the total row sum of the incidence structure of a Ryser design is equal to the totalcolumn sum, we have e r + e r = kα + 2 λβ . From equation (5) we know that e r + e r = λ ( v −
1) + r r .Hence kα + 2 λβ = λ ( v −
1) + r r which obtains,(8) r r = ( k − λ ) α + λ ( β + 1)Now using (8) in (7) we have αk − λ + β + 1 λ = ( v − ( k − λ ) α + λ ( β + 1) − v .Simplification then yields, α ( k − λ ) − α ( k − λ )[ v + ( k − λ )( v + 1)] + ( k − λ ) v ( v + 1 − λ ) = 0.Let(9) P ( α ) = α ( k − λ ) − α ( k − λ )[ v + ( k − λ )( v + 1)] + ( k − λ ) v ( v + 1 − λ ) . We claim that α = v and α = 1 are the only roots to the quadratic P ( α ).We have, P ( v ) = v ( k − λ ) − v ( k − λ )[ v + ( k − λ )( v + 1)] + ( k − λ ) v ( v + 1 − λ ).Simplification then obtains, P ( v ) = v [ − k ( k −
1) + λ ( v − α = v , thenthe Ryser design in consideration with two block sizes becomes a symmetric design ( v, k, λ )and hence it satisfies the relation k ( k −
1) = λ ( v − P (1) = ( k − λ ) − ( k − λ )[ v + ( k − λ )( v + 1)] + ( k − λ ) v ( v + 1 − λ ).Simplification then obtains, P (1) = v [ k ( v − k ) − λ ( v − α = 1, then the Ryser design in consideration with two block sizes is of Type-1. Let thisRyser design be derived from a symmetric design ( v, k, λ ′ ). Then k ( k −
1) = λ ′ ( v −
1) andin the Ryser design λ = k − λ ′ therefore k − k = λ ′ v − λ ′ ⇒ k = ( k − λ ) v + λ which givesus k ( v − k ) = λ ( v − α = 1 is a root of the quadratic (9).Successively differentiating quadratic (9) obtains P ′′ ( α ) = 2( k − λ ) > < α < v because k = 2 λ . Therefore P ( α ) does not change sign in the interval (1 , v ) and hence doesnot have any root in the interval (1 , v ). This proves that there is only one Ryser design withtwo block sizes in which one of the block sizes is 2 λ . (cid:3) By Theorem 2.8 all the Ryser designs in the same equivalence class have same value of r and r and hence have same value of ρ . Understanding the importance of the situation oftwo block sizes with 2 λ is helped by the following crucial hypothesis:Hypothesis H : Given any Ryser design E , no design in the equivalence class ( E ) of E haveboth a large and a small block, that is a large and a small block do not coexist in E or inany design equivalent to E . Theorem 3.4.
Assuming the hypothesis H (to hold for all the Ryser designs), every Ryserdesign is of Type-1 and thus hypothesis H implies the validity of the Ryser design conjecture.Proof. Let us assume that small and large blocks do not coexist in any equivalence class of aRyser design. Consider a Ryser design D of order v and index λ that does not have a small OME RESULTS ON THE RYSER DESIGN CONJECTURE-III 9 block. If D has at least two large blocks of different sizes say B , B with | B | > | B | > λ and if we complement this design with respect to block B . Then by Theorem 3.1 we have anew design D = D ∗ B in which B is a large block and B is a small block, a contradiction.Therefore there can not exist two different block sizes of large blocks in the original design D . Hence D is a Ryser design of order v and index λ with two block sizes with one blocksize 2 λ . Then by Theorem 3.3 D is of Type-1. (cid:3) Corollary 3.5.
Ryser design conjecture is equivalent to any one of the following statements:(i) Large and small blocks do not coexist in any Ryser design.(ii) There are no two different large (respectively small) block sizes in any Ryser design.(iii) There are exactly two block sizes in any Ryser design with one block size average.
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