1. Nasrin Soltankhah Department of Mathematical SciencesDepartment of Mathematical Sciences Alzahra University Tehran, I.R. IranTehran, I.R. Iran

2. Given a set of v treatments V. Let k and t be two positive integers such that t<k<v.

4. In a (v,k,t) trade both collections of blocks must cover the same set of elements. This set of elements is called the foundation of the trade and is denoted by found(T).

5. Example A (6,4,1) trade of volume 2 xy12 xy34 xy13 xy24 x12 x34 y13 y24 z14 z23 x13 x24 y14 y23 z12 z34 A (7,3,2) trade of volume 6

6. 1.Hedayat introduced the concept of trade [1] in the 1960s. 2.Hedayat and Li applied the method of trade-off and trades for building BIBDs by repeated blocks (1979-1980). 3.Milici and Quattrocchi introduced the steiner trade named it DMB (1984). 4. Hwang (1986), Mahmoodian and Soltankhah [1992 ] and Asgari and Soltankhah [ 2009] deal with the existence and non-existence of (v,k,t) trades.

7. Some Known results

8. xxxx xxxx

10. minimal Mimimal (v,k,t) trade has unique structure If found(T)=k+t+1

11. There exists (v,k,t) trade of volume m for

12. Combinatorial trade 1.Trade in other block designs 2.Trade in Latin squares (Latin trade) 3.G-trade in graphs (Decomposition H)

13. trade Latin trade -(v,k,t) Latin trade - (v,k,t) trade A Generalization of combinatorial trade

14. 1 2 3 4 3 4 2 1 4 3 1 2 2 1 4 3

15. Definition:

16. Example:

18. Definition

19. Example: x12 x34 y13 y24 z14 z23 x14 x23 y12 y34 z13 z24 x13 x24 y14 y23 z12 z34 3-way (7,3,2) trade xy12 xy34 xy13 xy24 xy14 xy23 3-way (6,4,1) trade

20. 123 147 158 248 267 357 368 456 124 138 157 237 268 467 458 356 127 135 148 246 238 367 457 568 3-way (8,3,2) trade

21. Application of Trade 1. Intersection problem 2. Defining set

23. BIBD Balanced incomplete block designs Let v, k, and λ be positive integers such that v > k ≥ 2. A (v, k, λ)-balanced incomplete block design ((v, k, λ)-BIBD) is a pair (X,A) such that the following properties are satisfied: 1. |X| = v, 2. each block contains exactly k points, and 3. every pair of distinct points is contained in exactly λ blocks.

24. A Steiner triple system of order v, or STS(v), is a (v, 3, 1)-BIBD. x12 x34 y13 y24 z14 z23 x14 x23 y12 y34 z13 z24 x13 x24 Y14 y23 z12 z34 x12 x34 y13 y24 z14 z23 xyz

26. x12 x34 y13 y24 z14 z23 xyz x14 x23 y12 y34 z13 z24 xyz x13 x24 y14 y23 z12 z34 xyz

28. Given parameters k, t. For which volume does there exist a µ – way (v, k, t) trade ? What is the volume spectrum ?

29. µ = 3

31. of volume m Construction 1

32. 12 34 13 24 14 23 3-way (4,2,1) trade Example: x12 x34 y13 y24 z14 z23 x13 x24 y14 y23 z12 z34 x14 x23 y12 y34 z13 z24 3-way (7,3,2) trade Unique structure

33. Construction 2

34. Example: 12 34 13 24 14 23 3-way (4,2,1) trade of volume 2 3-way (8,4,3) trade of volume 12

36. Construction 2

37. t m Construction 1 2 2 6 1 3 12 2 4 36 1 5 72 2

38. Question 1 Does there exist a 3-way (v,k,t) trade of volume less than Conjecture: The minimum volume is For t=2 For t=3

39. For t=2 and k=3 For t=3 and k=4 Question 2

41. k = t+1