Coloured and Directed Designs

For positive integers n≥k≥t, a collection MJX-tex-caligraphicscriptB of k‐subsets of an n‐set X is called a t‐packing if every t‐subset of X appears in at most one set in MJX-tex-caligraphicscriptB. In this paper, we investigate the existence of the maximum 3‐packings whenever n is sufficiently larger than k. When n≢20.3em(mod0.3emk−2), the optimal value for the size of a 3‐packing is settled. In other cases, lower and upper bounds are obtained where mostly differ by an additive constant depending only on k but one case that they differ by a linear bound in n.

“…Keevash's decomposition theorem can also be applied to decompose multigraphs and multihypergraphs in certain conditions. The following is a corollary of Theorem 6.5 in [16].…”

Section: Preliminariesmentioning

confidence: 80%

“…The following theorem is the minimum degree version of Keevash's decomposition theorem which can decompose

t

‐graphs with the large minimum degree (see Theorem 1.4 in [16] and also see Theorem 1.1 in [10]).…”

Section: Preliminariesmentioning

confidence: 99%

“…Now, we go through the case that

(n - 2) \equiv 0 (\mod k - 2)

and

(n - 1) (n - 2) \equiv 0 (\mod (k - 1) (k - 2))

. If, in addition,

n (n - 1) (n - 2) \equiv 0 (\mod k (k - 1) (k - 2))

, then the necessary conditions for the existence of a design hold and by the result of Keevash [16], we have

D (n, k, 3) = J (n, k, 3)

, for sufficiently large

n

. In the following theorem, we look into the other case when

n (n - 1) (n - 2) ≢ 0 (\mod k (k - 1) (k - 2))

.…”

Section: The Case (N−2)≡0(modk−2) and (N−1)(n−2)≡0(mod(k−1)(k−2))mentioning

confidence: 99%

“…If each

t

‐subset of

X

appears in exactly

λ

blocks, then it is called a

t - (n, k, λ)

design . Keevash [16], in a seminal work, establishing a well‐known long‐standing conjecture, proved that the obvious necessary conditions for the existence of

t

‐designs are also sufficient when

n

is sufficiently large with respect to

k

and

λ

. When these conditions are not met, we may ask for the maximum size of a

t

‐packing.…”

Section: Introductionmentioning

confidence: 99%

t

‐designs. In fact, the main result of Keevash [16], where the existence of

t

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Packing cliques in 3‐uniform hypergraphs

Javadi

Poorhadi

Fallah

2020

On a relation between bipartite biregular cages, block designs and generalized polygons

Araujo‐Pardo

Jajcay

Ramos-Rivera

et al. 2022

A bipartite biregular (m,n;g) $(m,n;g)$‐graph normalΓ ${\rm{\Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} $\{m,n\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$‐graph of minimum order. In their 2019 paper, Filipovski, Ramos‐Rivera, and Jajcay present lower bounds on the orders of bipartite biregular (m,n;g) $(m,n;g)$‐graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well‐known classical results relating the existence of k $k$‐regular Moore graphs of even girths g=6,8 $g=6,8$, and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an S(2,k,v) $S(2,k,v)$‐Steiner system yields the existence of a bipartite biregular k,v−1k−1;6 $\left(k,\frac{v-1}{k-1};6\right)$‐cage, and, vice versa, the existence of a bipartite biregular (k,n;6) $(k,n;6)$‐cage whose order is equal to one of our lower bounds yields the existence of an S(2,k,1+n(k−1)) $S(2,k,1+n(k-1))$‐Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of (3,n;6) $(3,n;6)$‐bipartite biregular cages for all integers n≥4 $n\ge 4$. Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of (n+1,n2+1;8) $(n+1,{n}^{2}+1;8)$‐, (n2+1,n3+1;8) $({n}^{2}+1,{n}^{3}+1;8)$‐, (n,n+2;8) $(n,n+2;8)$‐, (n+1,n3+1;12) $(n+1,{n}^{3}+1;12)$‐ and (n+1,n2+1;16) $(n+1,{n}^{2}+1;16)$‐bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14.

An explicit construction for large sets of infinite dimensional q $q$‐Steiner systems

Hawtin

2024

Let be a vector space over the finite field . A ‐Steiner system, or an , is a collection of ‐dimensional subspaces of such that every ‐dimensional subspace of is contained in a unique element of . A large set of ‐Steiner systems, or an , is a partition of the ‐dimensional subspaces of into systems. In the case that has infinite dimension, the existence of an for all finite with was shown abstractly by Cameron in 1995. This paper provides an explicit construction of an for all prime powers , all positive integers , and where has countably infinite dimension.