Candelabra Quadruple Systems And 3BD Closed Sets

Combinatorial structures called candelabra t-designs (CS(t, K, v)) can be used in recursiveconstructions to build other combinatorial structures of various properties. In this paper, we concentrate on the candelabra 3-designs with block size 4, which is usually called a candelabraquadruple system and denoted by CQS(gⁿ : s). Candelabra system was first introduced by Hanani. Mills, Hartman, Lenz and Granville all discussed the design CQS during their study of t-wise balanced design. Hartman and Phelps summarized the previous results on CQS in their paper "Steiner Quadruple Systems" and posed an open problem, which is to determine necessary and sufficient conditions for the existence of CQS(gⁿ : s). We will partially settle this problem in this paper.3BD closed set is a very powerful tool in the discussion of CQS, and 3BD on itself is also a very important topic of combinatorial designs. The known results of 3BD closed set are very limited and the main results are B₃({4}), B₃({4,6}), which were got by Hanani, and B₃({4,5}),B₃({4,5,6}), which were got by Ji. In this paper, we get a finite generating set for 3BD closed set K₇ and K₈ respectively. And discuss one of the remaining two parts of B₃({4,5,7}). We provide an approach to prove the existence of S(3, {4,5,7}, 12k + 7) by using the fundamental construction for 3BDs by A. Hartman.The paper is described as below:In chapter 1 we introduce the background and describe the concept of candelabra quadruplesystem and 3BD closed set. We also summarize others' work on CQS and list our main results.In chapter 2 we introduce the fundamental constructions for candelabra 3-designs and s-fan GDDs. We also get a construction for candelabra 3-design from s-fan GDD and a result of s-fan GDD.In chapter 3 We introduce some necessary conditions for the existence of a CQS(gⁿ : s)and settle existence when n = 4,5 and g is even. We get that the necessary conditions are also sufficient for n = 4; and for n = 5 and g≡0 (mod 2), the necessary conditions are also sufficient; and finally we get that there exists a CQS(gⁿ : s) for any n e {n≥3 : n(?)2,6 (mod 12) and n≠8}, where g≡0 (mod 6), s≡0 (mod 2) and 0≤s≤g. In this process, we completed the existence spectrum for G-designs with block size 4, that is, there exists a CQS(gⁿ : 0) if and only if n≡0 (mod 3) and g≡0 (mod 6), or n≡1,2 (mod 3) and g≡0 (mod 2).In chapter 4 we discuss the finite generating sets of 3BD closed sets K₇ and K₈, we get that:There exists an S(3, K, v) for any v≥7, where K = {7,8,…, 48,51,…, 55,59,60, 61,62,66,…,70,83,84,…,95,123}.There exists an S(3, K, v) for any v≥8, where K = {8,9,…, 49,51,…63,66,…, 71,75,…, 79,83,…, 97,104,123,…, 127,171,…, 183}.In chapter 5, we first introduce a recursive construction for S(3, {4,5,7}, 12k + 7). And discuss some small designs. Finally, we get that suppose there exist a CS(3, {4,5,7}, 127) of type (12¹⁰ : 7), an S(3, {4,5,7}, 79) and an S(3, {4,5,7}, 115), then there exists an S(3, {4,5,7}, 12k + 7) for any k∈{k is a non-negative integer }\{1,17,26,27,29,31,33}.In chapter 6, we list some further research problems.