Almost all Steiner triple systems have perfect matchings
نویسندگان
چکیده
منابع مشابه
Almost all Steiner triple systems have perfect matchings
We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a un...
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We use a free construction to prove the existence of perfect Steiner triple systems on a countably infinite point set. We use a specific countably infinite family of partial Steiner triple systems to start the construction, thus yielding 2א0 non-isomorphic perfect systems.
متن کاملSome new perfect Steiner triple systems
In a Steiner triple system STS(v)=(V,B), for each pair {a, b} ⊂ V , the cycle graph Ga,b can be defined as follows. The vertices of Ga,b are V \{a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25 and 33. We ...
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A triple system is cancellative if no three of its distinct edges satisfy A ∪ B = A ∪ C. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is cancellative. We prove that almost all cancellative triple systems with vertex set [n] are tripartite. This sharpens a theorem of N...
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Steiner triple systems are known to exist for orders n ≡ 1, 3 mod 6, the admissible orders. There are many known constructions for infinite classes of Steiner triple systems. However, Steiner triple systems that lack prescribed configurations are harder to find. This paper gives a proof that the spectrum of orders of 5-sparse Steiner triple systems has arithmetic density 1 as compared to the ad...
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ژورنال
عنوان ژورنال: Proceedings of the London Mathematical Society
سال: 2020
ISSN: 0024-6115,1460-244X
DOI: 10.1112/plms.12373