Subdesigns in Steiner quadruple systems (original) (raw)

Abstract

Steiner quadruple system of order v, denoted SQS(v), is a pair (X, a), where X is a set of cardinality v, and 1 is a set of 4-subsets of C (called blocks), with the property that any 3-subset of X is contained in a unique block. If (X, 0) is an SQS(v) and (Y, C) is an SQS(w) with YE X and Cc@, we say that (Y, C) is a subdesign of (X, 0). Hanani has shown that an SQS(v) exists for all v E 2 or 4 (mod 6) and when v E {0, 1); such integers v are said to be admissible. A necessary condition for the existence of an SQS(v) with a subdesign of order w is that v = w or v > 2w. In this paper we show the existence of an explicitly computable constant k (independent of w) such that for all admissible v and all admissible w with v > kw there exists an SQS(v) containing a subdesign of order w. We also show that for any sufficiently large w we can take k = 12.54. To establish these results we introduce several new constructions for SQS, and we also consider the subdesign problem for related classes of designs. 0 1991 Academic press, h. 1. INTR~D~JcTI~N This paper is concerned with the existence problem for SQS(v) with a subdesignoforder w. LetA,=(O,l}u{u~2:~~2or4(mod6)) denote the set of admissible integers. For w E A,, we define S, to be the set of all orders u for which there exists an SQS(u) with a subdesign of order w. Define s, to be the least admissible integer vO, such that for all u EAT, u 2 u0 we have u E S,, if such a u0 exists; otherwise s, = co. In Section 3 we shall show that s, is finite for all w E A,, and then in Section 4 we shall show that s, < kw for some absolute constant k. In Section 5 we improve the constant k. To do this we need constructions for SQS(u) with subdesigns. These constructions are described in Section 2, including a review of existing constructions as applied to the subsystem problem. Several new constructions are also described in Section 2, and the details of these new results are given in Section 6. In Section 7 we give some generalizations of our results and, in Section 8, pose some problems.

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References (50)

  1. u,,, b,, c2]: a + b + c = ei (mod 2f), (a, b, c) = (1, 1,O) (mod 2)) {CcoZi+19
  2. a,,, b,, c,]: a + b + c = di (mod 2f), (a, b, c) = (0, 0, 1) (mod 2))
  3. C032i+17 cc,,b,,c,]:u+b+c~di+1(mod2f),(u,b,c)~(0,1,1)(mod2)} {Coo*i+l~ u,,b,,c,]:u+b+c~e,+1(mod2f),(u,b,c)~(1,0,0)(mod2)}
  4. C~*;+1, %Y b,, c3]: a + b + c = ei (mod 2f), (a, b, c) = (0, 0,O) (mod 2)) {CC02i+lr
  5. u,,b,,c,]:u+b+c=di(mod2f),(u,b,c)-(l,l,l)(mod2)}
  6. W2i+ Ir a19 b,,c,]:u+b+c-di+l (mod2f),(u,b,c)~(1,l,O)(mod2)} {c~*i+l? a13 b,,c3]:u+b+c-ei+l (mod2f),(u,b,c)-(O,O,l)(mod2)}.
  7. Note that each of these sets of blocks contains f' distinct blocks. Every 3-subset of x' of the form cojxk y, with k # m, is contained in precisely one block. This can be checked methodically by considering the six possibilities for {k, m} and the eight possible parities of (j, x, y). The other 3-subsets of X' contained in the above blocks are those of the forms: u,b,c,withu+b+c~DuE,andu=b(mod2) u,b,c,withu+b+c~(DuE)+landufb(mod2) u,b,c,witha+b+cEDuE,andb=c(mod2) u,b,c,withu+b+cE(DuE)+landbfc(mod2).
  8. We now form the following sets of blocks in 99'. {C~,,b,,g,,~,l:~~b(mod2),{ g,g}EGC,u+b+2c=O(mod2f)} {Cao,b,,g3,g31:afb(mod2), { g, g}EG,,u+b+2c=l (mod2f)j {[g,,g,,b2,c3]:b~c(mod2), {g,g}EG,,2u+b+c-O(mod2f))
  9. g,,g,,b~,c,]:bfc(mod2),{g,g}~G,,2u+b+c~l(mod2f)}. Each of these sets of blocks contains 2j-'(f-s) distinct blocks. The 3-subsets of X' contained in these blocks include those of the forms u,b,g, with u+b+g#DuE, anduEb(mod2).
  10. Since G, is a partition of Zzf\( (D u E) + 2c), we see that g $ (D u E) + 2~ and therefore u+b+g$(DuE)+2c+u+b=DuE.
  11. u,b,g3withu+b+g#(DuE)+1,andufb(mod2) g,b2c3 with g + b + c $ (D u E), and b = c (mod 2)
  12. All triples of the form co,a]b, with j# k occur precisely once in either the A fragment or the B fragment, but not both. Specifically A contains all those with i < n, B contains all those with i > n. (3) All triples of the form ai(a + d)ib, with i# j, do { 1,2,3) occur precisely once in either the A or the B fragment, but not both. Specifically n=2,4: A contains d = 2, 3; B contains d = 1.
  13. n = 6, 8, 10, 12: A contains none; B contains all. To simplify the construction we give below a list of some useful one- factors of the graph with vertex set 2,.
  14. F,: co, 11c2,31c4,51
  15. F,: co, 5m21c3,41 6: ~0,31c2,41c1,51 F4: [L 41c0,21c3,51
  16. F,: CT 51CO,4lCL 31 ~6 co, 31c1,41c2,51.
  17. DFA(2)( =Design 39 of [
  18. Cai, (a + 2)i, (a + 3b + 1 )i+ 1, (a+36+ l)i+Z]:aeZg, iEZ3, be (0, l} [ai,(a+2)i,(a+3)i+,,(a+5),-,]:aEZg,iEZ3,kE{1,2}
  19. Cai,(a+2)i,ai+l,(a+2)i+,1:aEZ,,iEZ, [coi,a,,b,,c,]:a+b+cz3j(mod6),a,b,cEZg, je{O,l} Cai, bi, Ci+ I 9 di+,]:ieZ3, [a,b]EFe, [c,d]EFs. DFB(2)( =Design 40 of [
  20. = DFB(4) Caj, (a+ lh, bi+l, ci+z]:a+b+cr2i(mod6),ieZ3,a,b,cEZg. DFA(4)( =Design l.B of [12]) [ai, (a+3b)i+l, (1-2a-3b)i+2, (5-2a-3b)i+,]: aeZ6, iEZx, be (0, l}
  21. Cai, (a + 2)i, (a + 3b)i+ 1, (a+3b+2),+,]:aEZ6, ieZ3, be (0, I} [ooj, a,, b,, cz]: a + b + c = k (mod 6), a, b, CE&, CL k)E {(O,O)(L 2)(2, 3)(3,4)}
  22. Cai, bi, ci+ 1, di+,l:iEZs, Ca,blEF6, Cc,dlEFe DFA(6)( gIY(
  23. = DFA(8) = DFA( 10) = DFA( 12) [coj,a,,b,,c,]:a+b+c=j(mod6),a,b,c,j~Zg DFB(6)( EG(
  24. Carr (a+l)i,bi+l,C;+Z ]:a+b+c-2i(mod6),iEZ3,a,b,cEZ6 [ai, bi, c;+ 19 cI~+~]:~EZ,, [u,b]EFk, [c,d]~F~,k~{3,4, 5}. DFB(8) First three classes of blocks in DFA(2), together with [co,,uO,b,,c,]:a+b+c-3j(mod6),u,b,cEZg,jE{6,7}
  25. Cuitbi,Ci+l~ d,+,]:i~Z~,[u,b]~F~,[c,d]~F~,k~{l,2,6}. DFB( 10) First two classes of blocks in DFA(4), together with [coj,u,,b,,c,]:a+b+c=k(mod6), a, b, c E Z6, (j, k) E { (6,0)(7, 2)(8, 3)(9,4))
  26. ui, bi, ci+ 1, di+I]:i~Z3, [a,b]EFk, [c,d]~F~,k~{1,2,6}.
  27. DFB( 12) [coi,u,,b,,c,]:u+b+c~j(mod6),j~{6,7 ,..., 11)
  28. lui, bi, Ci+ I 3 di+,]:iEZ,,[u,b]Elrk,[c,d]EFk,kE{1,2,3,4,5}. The other ingredients in the hextupling constructions are H(24), H(36) (see Hanani [7] or Mills [ 17]), and the following designs with subdesigns. LEMMA 6.2. Designs which validate the following assertions exist. {12+n,24+n}~S, for n~{2,4,8,10)
  29. 12+n, 24+n} GG, for ne (6, 12).
  30. Proof: The twelve designs are easily constructed using the doubling and tripling constructions. We now describe the hextupling construction. Let (Xu {A, B), g) be an S(v) if v = 2 or 4 (mod 6), and let it be a G(v) when o=O (mod 6), with both A and B in the same block of size 6. We shall construct an S(6(u -2) + n), respectively G(6(v -2) + n), when n=2,4,8, 10, respectively, n=6, 12. Let Zn={coO, co1 ,..., co,-,}. The point set of the new design will be (Z, x X) u Z,. The blocks are con- structed as follows. For each block in SY containing both A and B, say REFERENCES
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