On Kaleidoscope Designs Francesca Merola Roma Tre University Joint - PowerPoint PPT Presentation

On Kaleidoscope Designs Francesca Merola Roma Tre University Joint work with Marco Buratti

notation � V � ( v , k , λ )-design: | V | = v set of points, B set of blocks, B ⊂ , k |B| = b , such that any two points belong to exactly λ blocks x 5 x 4 x 2 x 6 x 0 x 1 x 3 (9 , 3 , 1)-design (7 , 3 , 1)-design

definition • let D be a set of ( k , h , 1)-designs with b blocks • the points of the designs in D belong to a given v -set V • the b blocks of designs in D are colored with the same b different colors c 0 , c 1 ,. . . , c b − 1 . • We say that D is a Kaleidoscope Design of order v and type ( k , h , 1), briefly a ( k , h , 1) K ( v ) • if for any two distinct points x , y of V and any color c i there is exactly one design of D in which the block having color c i contains x and y . • in this talk we shall deal mostly with the case ( k , h , 1) = (7 , 3 , 1) - Fano case • and ( k , h , 1) = (9 , 3 , 1) - affine case

Fano case • a set D of (7 , 3 , 1)-designs – Fano Planes with points from a v -set V • the seven lines in each plane are coloured with seven different colours c 0 , c 1 , c 2 , c 3 , c 4 , c 5 , c 6 . • for each pair of points x , y in V and each colour c i • ∃ ! plane in D in which the line of colour c i contains x and y

affine case • a set D of (9 , 3 , 1)-designs with points from a v -set V • the 12 lines in each plane are coloured with 12 different colours c 0 , c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , c 7 , c 8 , c 9 , c 10 , c 11 . • for each pair of points x , y in V and each colour c i • ∃ ! design in D in which the line of colour c i contains x and y

necessary conditions • the underlying - uncoloured - structure is a ( v , k , b ) design • each pair of points x and y ∈ V appear together in b “small” designs • so the usual admissibility conditions for designs apply • for the (7 , 3 , 1) case, we have v ≡ 1 (mod 6) • for the (9 , 3 , 1) case, we have v ≡ 1 , 3 (mod 6) • for ( k , h , 1) K ( v ) we have v ( v − 1) h ( h − 1) ∈ Z and k ( v − 1) h ( h − 1) ∈ Z

coloured designs • a Kaleidoscope Design is a special instance of the very broad notion of coloured design/edge-coloured graph decomposition • various definitions with more/less generality • important asymptotic result by Lamken and Wilson (2000) • studied by Colbourn and Stinson (1988); Caro, Roditty and Sch¨ onheim (1995,1997,2002); Adams, Bryant and Jordon (2006) • many concepts (e.g. perfect cycle systems, whist tournaments, nested triple systems, . . . ) can be rephrased in this setting

repetitions • if a ( v , k , 1) design exists, by replicating each block b times and shifting cyclically the colours one has a Kaleidoscope design • for instance in the Fano case for v=7, we just repeat the one block (0 , 1 , 2 , 3 , 4 , 5 , 6) seven times { 0 , 1 , 3 } , { 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 0 } , { 5 , 6 , 1 } , { 6,0,2 } { 0 , 1 , 3 } , { 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 0 } , { 5 , 6 , 1 } , { 6,0,2 } { 0 , 1 , 3 } , { 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 0 } , { 5 , 6 , 1 } , { 6,0,2 } { 0 , 1 , 3 } , { 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 0 } , { 5 , 6 , 1 } , { 6,0,2 } { 0 , 1 , 3 } , { 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 0 } , { 5 , 6 , 1 } , { 6,0,2 } { 0 , 1 , 3 } , { 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 0 } , { 5 , 6 , 1 } , { 6,0,2 } { 0 , 1 , 3 } , { 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 0 } , { 5 , 6 , 1 } , { 6,0,2 }

example - a (7 , 3 , 1) K (19) 8 2 5 11 0 1 4 • in Z 19 consider the 7-ple B = (0 , 1 , 2 , 4 , 5 , 11 , 8) • as blocks take B = { mB + k , m ∈ { 1 , 2 6 , 2 12 } , k ∈ Z 19 } = { (0 , 1 , 2 , 4 , 5 , 11 , 8) + k , (0 , 7 , 14 , 9 , 16 , 1 , 18) + k , (0 , 11 , 3 , 6 , 17 , 7 , 12) + k ; k ∈ Z 19 } • we have a (7 , 3 , 1) K (19) regular under Z 19 • taking as lines the triples in position i , i + 1 , i + 3 (ind mod 7) and colouring the i -th triple with the i -th colour • so for B we have { 0 , 1 , 4 } , { 1 , 2 , 5 } , { 2 , 4 , 11 } , { 4 , 5 , 8 } , { 5 , 11 , 0 } , { 11 , 8 , 1 } , { 8 , 0 , 2 }

example - a (9 , 3 , 1) K (19) • in Z 19 consider the 9-ple B = (0 , 1 , 2 , 3 , 7 , 16 , 8 , 4 , 10) = ( b ∞ , b 0 , b 1 , b 2 , b 3 , b 4 , b 5 , b 6 , b 7 ) • as blocks take B = { mB + k , m ∈ { 1 , 2 6 , 2 12 } , k ∈ Z 19 } = { (0 , 1 , 2 , 3 , 7 , 16 , 8 , 4 , 10) + k , (0 , 7 , 14 , 2 , 11 , 17 , 18 , 9 , 13) + k , (0 , 11 , 3 , 14 , 1 , 5 , 12 , 6 , 15) + k ; k ∈ Z 19 } • we have a (9 , 3 , 1) K (19) regular under Z 19 • taking as lines the triples { b ∞ , b 0 , b 4 } , { b ∞ , b 1 , b 5 } , { b ∞ , b 2 , b 6 } , { b ∞ , b 3 , b 7 } and the 8 lines of the form { b i , b i +1 , b i +3 } (ind mod 8) • and colouring each of these 12 lines with a different color { 0 , 1 , 16 } , { 0 , 2 , 8 } , { 0 , 3 , 4 } , { 0 , 7 , 10 } , { 1 , 2 , 7 } , { 2 , 3 , 16 } , { 3 , 7 , 8 } , { 7 , 16 , 4 } , { 16 , 0 , 10 } , { 8 , 4 , 1 } , { 4 , 10 , 2 } , { 10 , 1 , 3 }

existence results for the Fano case • we used difference methods, and considered the case in which v is prime or a prime power Theorem A regular (7 , 3 , 1) K ( v ) exists for all v ≡ 1 (mod 6) , whenever v is prime or a prime power, v � = 13 . • look for a starting block ( b 0 , b 1 , b 2 , b 3 , b 4 , b 5 , b 6 ) such that the differences in the triples { b 0 , b 1 , b 3 } , { b 1 , b 2 , b 4 } , . . . , { b 6 , b 0 , b 2 } satisfy specific cyclotomic conditions • a result of Buratti and Pasotti (2009) guarantees the existence of such a block for v > a bound β ≈ 4 800 000 • we explicitly built a design for all values of v smaller than β , and found that no regular design exists for v = 13

recursive existence results • this result can be applied recursively with the help of difference matrices to give Theorem A regular (7 , 3 , 1) K ( v ) exists for all v whose prime power factors are all ≡ 1 (mod 6) (but not = 13)

PBDs • a ( v , K , λ )- Pairwise Balanced Design , or ( v , K , λ )-PBD • is a pair ( V , B ), where V is a v -set of points and B is a set of blocks, subsets of V with | B | ∈ K ∀ B ∈ B • s.t. each pair of points of V is contained in λ blocks B ∈ B • when K = { k } , this gives a ( v , k , λ )-design. • PBDs are very useful in constructing “ordinary” designs • and also in building Kaleidoscope designs: • the existence of a ( v , K , 1)-PBD together with the existence of a ( k , h , 1) K ( w ) for all w ∈ K implies the existence of a ( k , h , 1) K ( v )

PBDs existence results • we can make use of the following result on PBDs Theorem (Mullin and Stinson (1987), Greig (1999)) Let K be the set of prime powers ≡ 1 (mod 6) ; then, for all v ≡ 1 (mod 6) with at most 22 exceptions, a ( v , K , 1) − PBD exists. • a (7 , 3 , 1) K ( v ) can exist only if v ≡ 1 (mod 6), and we have proved existence for all for all orders that are prime powers ≡ 1 (mod 6) (except for 13!!!) • if we build a (7 , 3 , 1) K (13), this result together with our existence result implies the existence of a (7 , 3 , 1) K ( v ) for all admissible values with at most 22 exceptions • even without a (7 , 3 , 1) K (13), the result of Mullin and Stinson allows us to prove the existence of a (7 , 3 , 1) K ( v ) for many non prime power values of v

(7 , 3 , 1) K (13) • the underlying, uncolored design is in this case a (13 , 7 , 7)-design • there are 19 072 802 such designs (Kaski and ¨ Osterg˚ ard (2004))

existence results for the affine and general case • we could use similar methods to obtain existence results in the affine case: via difference methods and the use of cyclotomic conditions, once more using Buratti and Pasotti (2009) we can state that Theorem A regular (9 , 3 , 1) K ( v ) exists for all v ≡ 1 (mod 6) , whenever v is prime or a prime power, v > than a bound β . • the bound in this case is β ≈ 809 000 000 • harder (but probably possible) to consider all the values smaller than β - once more, � ∃ regular for v = 13 • were we trying to apply the same methods to (13 , 3 , 1) K ( v ), the bound would be β ≈ 1 . 38391 · 10 13 • also we cannot rely on the results on PBDs; as in the Fano case, a non regular (9 , 3 , 1) K (13) might exist, but not a (9 , 3 , 1) K (7)!

a kaleidoscopic anniversary the kaleidoscope was invented 200 years ago by David Brewster (1781-1868), a Scottish physicist, mathematician, astronomer, inventor, writer, historian of science and university principal