Let K be an abelian group of order v. A Steiner quadruple system of order v (SQS(v)) (K,B) is called symmetric K-invariant if for each B∈B, it holds that B+x∈B for each x∈K and B=−B+y for some y∈K. ...When the Sylow 2-subgroup of K is cyclic, a necessary and sufficient condition for the existence of a symmetric K-invariant SQS(v) was given by Munemasa and Sawa, which is a generalization of a necessary and sufficient condition for the existence of a symmetric cyclic SQS(v) shown in Piotrowski's thesis in 1985. In this paper, we prove that a symmetric K-invariant SQS(v) exists if and only if v≡2,4(mod6), the order of each element of K is not divisible by 8, and there exists a symmetric cyclic SQS(2p) for any odd prime divisor p of v.
A set grading on the split simple Lie algebra of type D13, that cannot be realized as a group-grading, is constructed by splitting the set of positive roots into a disjoint union of pairs of ...orthogonal roots, following a pattern provided by the lines of the projective plane over GF(3). This answers in the negative 3, Question 1.11.
Similar non-group gradings are obtained for types Dn with n≡1(mod12), by substituting the lines in the projective plane by blocks of suitable Steiner systems.
Disjoint q-Steiner systems in dimension 13 Braun, Michael; Wassermann, Alfred
Electronic notes in discrete mathematics,
March 2018, 2018-03-00, Volume:
65
Journal Article
Open access
We report the computer construction of 1316 mutually disjoint 2-(13, 3, 1)2 subspace designs. By combining disjoint designs and using supplementary subspace designs we conclude that subspace designs ...exist for 1≤λ≤2047.
A linear hypergraph, also known as a partial Steiner system, is a collection of subsets of a set such that no two of the subsets have more than one element in common. Most studies of linear ...hypergraphs consider only the uniform case, in which all the subsets have the same size. In this paper we provide, for the first time, asymptotically precise estimates of the number of linear hypergraphs in the non-uniform case, as a function of the number of subsets of each size.
Ramsey theorem for designs Hubička, Jan; Nešetřil, Jaroslav
Electronic notes in discrete mathematics,
August 2017, 2017-08-00, Volume:
61
Journal Article
We prove that for every choice of parameters 2≤t≤k and 1≤λ the class PD→ktλ of linearly ordered partial designs with parameters k, t, λ is a Ramsey class. Thus, together with the recent spectacular ...results of Keevash, one obtains that the class of linearly ordered designs D→ktλ is a Ramsey class.
In this corrigendum, we correct the proof of Theorem 10 from our paper titled „Bounds on the number of edges of edge-minimal, edge-maximal and
-hypertrees”.
For an ordering of the blocks of a design, the point sum of an element is the sum of the indices of blocks containing that element. Block labelling for popularity asks for the point sums to be as ...equal as possible. For Steiner systems of order
v
and strength
t
in general, the average point sum is
O
(
v
2
t
-
1
)
; under various restrictions on block partitions of the Steiner system, the difference between the largest and smallest point sums is shown to be
O
(
v
(
t
+
1
)
/
2
log
v
)
. Indeed for Steiner triple systems, direct and recursive constructions are given to establish that systems exist with all point sums equal for more than two thirds of the admissible orders.
Liu et al. (IEEE Trans Inf Theory 68:3096–3107, 2022) investigated a class of BCH codes
C
(
q
,
q
+
1
,
δ
,
1
)
with
q
=
δ
m
a prime power and proved that the set
B
δ
+
1
of supports of the minimum ...weight codewords supports a Steiner system
S
(
3
,
δ
+
1
,
q
+
1
)
. In this paper, we give an equivalent formulation of
B
δ
+
1
in terms of elementary symmetric polynomials and then construct a number of mutually disjoint Steiner systems S
(
3
,
δ
+
1
,
δ
m
+
1
)
when
m
is even and a number of mutually disjoint G-designs G
(
δ
m
+
1
δ
+
1
,
δ
+
1
,
δ
+
1
,
3
)
when
m
is odd. In particular, the existence of three mutually disjoint Steiner systems
S
(
3
,
5
,
4
m
+
1
)
or three mutually disjoint G-designs G
(
4
m
+
1
5
,
5
,
5
,
3
)
is established.
Let
X
be a simplicial complex on vertex set
V
. We say that
X
is
d
-representable if it is isomorphic to the nerve of a family of convex sets in
R
d
. We define the
d
-boxicity of
X
as the minimal
...k
such that
X
can be written as the intersection of
k
d
-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of
X
is a set
τ
⊂
V
such that
τ
∉
X
but
σ
∈
X
for any
σ
⊊
τ
. We prove that the
d
-boxicity of a simplicial complex on
n
vertices without missing faces of dimension larger than
d
is at most
⌊
n
d
/
(
d
+
1
)
⌋
. The bound is sharp: the
d
-boxicity of a simplicial complex whose set of missing faces form a Steiner
(
d
,
d
+
1
,
n
)
-system is exactly
n
d
/
(
d
+
1
)
. One of the main ingredients in the proof is the following bound on the representability of a complex: let
V
1
,
…
,
V
k
be subsets of
V
such that
V
i
∉
X
for all
1
≤
i
≤
k
, and for any missing face
τ
of
X
there is some
1
≤
i
≤
k
satisfying
|
τ
\
V
i
|
≤
1
. Then,
X
can be written as an intersection
X
=
⋂
i
=
1
k
X
i
, where, for all
1
≤
i
≤
k
,
X
i
is a
(
|
V
i
|
-
1
)
-representable complex. In particular,
X
is
(
∑
i
=
1
k
(
|
V
i
|
-
1
)
)
-representable.
A
partial
(
n
,
k
,
t
)
λ
-system
is a pair
(
X
,
B
)
where
X
is an
n
-set of
vertices
and
B
is a collection of
k
-subsets of
X
called
blocks
such that each
t
-set of vertices is a subset of at most
...λ
blocks. A
sequencing
of such a system is a labelling of its vertices with distinct elements of
{
0
,
…
,
n
-
1
}
. A sequencing is
ℓ
-
block avoiding
or, more briefly,
ℓ
-good
if no block is contained in a set of
ℓ
vertices with consecutive labels. Here we give a short proof that, for fixed
k
,
t
and
λ
, any partial
(
n
,
k
,
t
)
λ
-system has an
ℓ
-good sequencing for some
ℓ
=
Θ
(
n
1
/
t
)
as
n
becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case
k
=
t
+
1
where results of Kostochka, Mubayi and Verstraëte show that the value of
ℓ
cannot be increased beyond
Θ
(
(
n
log
n
)
1
/
t
)
. A special case of our result shows that every partial Steiner triple system (partial
(
n
,
3
,
2
)
1
-system) has an
ℓ
-good sequencing for each positive integer
ℓ
⩽
0.0908
n
1
/
2
.
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