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This paper is a contribution to the classification of all pairs (T,G) $({\mathscr{T}},G)$, where T ${\mathscr{T}}$ is a triple system and G $G$ is a block‐transitive but not flag‐transitive automorphism group of T ${\mathscr{T}}$. We prove that if the socle of G $G$ is a sporadic or alternating group, then one of the following holds: (i)T ${\mathscr{T}}$ is a TS(10,2) $TS(10,2)$ and G≅A5 $G\cong {A}_{5}$;(ii)T ${\mathscr{T}}$ is a TS(10,4) $TS(10,4)$ and G≅S5 $G\cong {S}_{5}$;(iii)T ${\mathscr{T}}$ is a TS(55,28) $TS(55,28)$ and G≅A11 $G\cong {A}_{11}$ or S11 ${S}_{11}$;(iv)T ${\mathscr{T}}$ is a TS(55,λ) $TS(55,\lambda)$ with λ∈{4,8,16} $\lambda \in \{4,8,16\}$ and G≅M11 $G\cong {M}_{11}$. [ABSTRACT FROM AUTHOR]

We construct the combinatorial Pöppe triple system, or ternary algebra, that underlies the non-commutative nonlinear Schrödinger (NLS) and modified Korteweg–de Vries (mKdV) hierarchy. We demonstrate that the Pöppe triple system provides an effective and systematic procedure for establishing that the NLS and mKdV equations are directly linearisable, which, in principle, extends to the whole hierarchy. This naturally extends the combinatorial Pöppe algebra, recently used to constructively establish integrability and uniqueness for the whole non-commutative potential Korteweg–de Vries hierarchy, to the NLS and mKdV hierarchy.

For distributed storage based on data replication, an optimal data placement is vital to minimize the variance of the number of available files. Well-balanced triple systems and nearly well-balanced triple systems (NWBTSs) were proposed to produce optimal data placements for triple replication. This article concentrates on the existence of NWBTSs. By constructing candelabra systems with various desirable partitions, many new constructions for NWBTSs are developed. The main result of this paper is that there always exist optimal data placements for triple replication on a v-set for all positive integers v possibly except when v ≡ 4 (mod 24) or v = 50 , 74 . [ABSTRACT FROM AUTHOR]

Abstract In this article, we are pleased to investigate multiple positive solutions for a system of Hadamard fractional differential equations with ( p 1 , p 2 , p 3 ) $(p_{1}, p_{2}, p_{3})$ -Laplacian operator. The main results rely on the standard tools of different fixed point theorems. Finally, we demonstrate the application of the obtained results with the aid of examples.

A balanced incomplete block design is a set system in which all pairs of distinct elements occur with a constant frequency. By contrast, a Sarvate-Beam design induces an interval of distinct frequencies on pairs. In this paper, we settle the existence of a Sarvate-Beam variant of group divisible designs of uniform type with block size three. [ABSTRACT FROM AUTHOR]

In this article, we are pleased to investigate multiple positive solutions for a system of Hadamard fractional differential equations with (p 1 , p 2 , p 3) -Laplacian operator. The main results rely on the standard tools of different fixed point theorems. Finally, we demonstrate the application of the obtained results with the aid of examples. [ABSTRACT FROM AUTHOR]

This article aims to present a novel method, namely wheel partition technique, for constructing a new cyclic 12-fold triple system called cyclic triple factorization denoted by CTF(v). We prove the existence of CTF(v) for v = 12n + 10. Then, an arrangement of v x 2(v - 1) triples of CTF(v) is developed using the idea of decomposition of wheel graph into triangles (triples). Moreover, the starter triples algorithms of CTF(v) are formulated to generate all triples. [ABSTRACT FROM AUTHOR]

A TS (v , λ) is a pair (V , B) where V contains v points and B contains 3-element subsets of V so that each pair in V appears in exactly λ blocks. A 2-block intersection graph (2-BIG) of a TS (v , λ) is a graph where each vertex is represented by a block from the TS (v , λ) and each pair of blocks B i , B j ∈ B are joined by an edge if | B i ∩ B j | = 2 . We show that there exists a TS (v , λ) for v ≡ 0 or 4 (mod 12) whose 2-BIG is Hamiltonian for all admissible (v , λ) . This is equivalent to the existence of a TS (v , λ) with a cyclic 2-intersecting Gray code. [ABSTRACT FROM AUTHOR]

We prove that, for sufficiently large n, every graph of order n with minimum degree at least 0.852n has a fractional edge-decomposition into triangles. We do this by refining a method used by Dross [SIAM J. Discrete Math., 30 (2016), pp. 36--42] to establish a bound of 0.9n. By a result of Barber, Kuhn, Lo, and Osthus [Adv. Math., 288 (2016), pp. 337--385], our result implies that, for each epsilon > 0, every graph of sufficiently large order n with minimum degree at least (0.852 + epsilon)n has a triangle decomposition if and only if it has all even degrees and number of edges a multiple of three. [ABSTRACT FROM AUTHOR]

We reach beyond the celebrated theorems of Erdȍs-Ko-Rado and Hilton-Milner, and a recent theorem of Han-Kohayakawa, and determine all maximal intersecting triples systems. It turns out that for each \(n\geq 7\) there are exactly 15 pairwise non-isomorphic such systems (and 13 for \(n=6\)). We present our result in terms of a hierarchy of Turán numbers \(\operatorname{ex}^{(s)}(n; M_2^{3})\), \(s\geq 1\), where \(M_2^{3}\) is a pair of disjoint triples. Moreover, owing to our unified approach, we provide short proofs of the above mentioned results (for triple systems only). The triangle \(C_3\) is defined as \(C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\},\{x_2,y_1,x_3\}\}\). Along the way we show that the largest intersecting triple system \(H\) on \(n\geq 6\) vertices, which is not a star and is triangle-free, consists of \(\max\{10,n\}\) triples. This facilitates our main proof's philosophy which is to assume that \(H\) contains a copy of the triangle and analyze how the remaining edges of \(H\) intersect that copy.

A graph is n-existentially closed (n-e.c.) if for any disjoint subsets A, B of vertices with A ∪ B = n , there is a vertex z ∉ A ∪ B adjacent to every vertex of A and no vertex of B. For a block design with block set B , its block intersection graph is the graph whose vertex set is B and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be 2-e.c. In particular, we study the λ -fold triple systems with λ ≥ 2 and determine for which parameters their block intersection graphs are 1- or 2-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called { 1 } -block intersection graphs are investigated, and the necessary and sufficient conditions for such graphs to be 2-e.c. are established. [ABSTRACT FROM AUTHOR]

A Berge-K4 in a triple system is a configuration with four vertices v1, v2, v3, v4 and six distinct triples {eij: 1 ≤ i < j ≤ 4} such that{ui, vj}⊂ eij for every 1 ≤ i < j ≤ 4. We denote by B the set of Berge-K4 configurations. A triple system is B -free if it does not contain any member of B . We prove that the maximum number of triples in a B -free triple system on n ≥ 6 points is obtained by the balanced complete 3-partite triple system: all triples {abc: a ϵ A, b ϵ B, c ϵ C} where A,B,C is a partition of n points with [n/3]= |A| ≤|B| ≤ |C| = n/33 [ABSTRACT FROM AUTHOR]

Abstract: Given a combinatorial design D with block set B, the block‐intersection graph (BIG) of D is the graph that has B as its vertex set, where two vertices B 1 ∈ B and B 2 ∈ B are adjacent if and only if | B 1 ∩ B 2 | > 0. The i‐block‐intersection graph (i‐BIG) of D is the graph that has B as its vertex set, where two vertices B 1 ∈ B and B 2 ∈ B are adjacent if and only if | B 1 ∩ B 2 | = i. In this paper, several constructions are obtained that start with twofold triple systems (TTSs) with Hamiltonian 2‐BIGs and result in larger TTSs that also have Hamiltonian 2‐BIGs. These constructions collectively enable us to determine the complete spectrum of TTSs with Hamiltonian 2‐BIGs (equivalently TTSs with cyclic 2‐intersecting Gray codes) as well as the complete spectrum for TTSs with 2‐BIGs that have Hamilton paths (i.e. for TTSs with 2‐intersecting Gray codes). In order to prove these spectrum results, we sometimes require ingredient TTSs that have large partial parallel classes; we prove lower bounds on the sizes of partial parallel classes in arbitrary TTSs, and then construct larger TTSs with both cyclic 2‐intersecting Gray codes and parallel classes. [ABSTRACT FROM AUTHOR]