相关论文: Cancellative hypergraphs and Steiner triple system…
A triple system is cancellative if no three of its distinct edges satisfy $A \cup B=A \cup C$. 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…
Let $G$ be a cancellative $3$-uniform hypergraph in which the symmetric difference of any two edges is not contained in a third one. Equivalently, a $3$-uniform hypergraph $G$ is cancellative if and only if $G$ is $\{F_4, F_5\}$-free, where…
A Steiner triple system is a set $S$ together with a collection $\mathcal{B}$ of subsets of $S$ of size 3 such that any two elements of $S$ belong to exactly one element of $\mathcal{B}$. It is well known that the class of finite Steiner…
In this paper, we initiate the study of discrepancy questions for combinatorial designs. Specifically, we show that, for every fixed $r\ge 3$ and $n\equiv 1,3 \pmod{6}$, any $r$-colouring of the triples on $[n]$ admits a Steiner triple…
For various triple systems $F$, we give tight lower bounds on the number of copies of $F$ in a triple system with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of…
For various quadruple systems F, we give asymptotically sharp lower bounds on the number of copies of F in a quadruple system with a prescribed number of vertices and edges. Our results extend those of Furedi, Keevash, Pikhurko, Simonovits…
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…
For every positive integer $t$ we construct a finite family of triple systems ${\mathcal M}_t$, determine its Tur\'{a}n number, and show that there are $t$ extremal ${\mathcal M}_t$-free configurations that are far from each other in…
We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper…
We establish an upper bound on the minimum codegree necessary for the existence of spanning, fractional Steiner triple systems in $3$-uniform hypergraphs. This improves upon a result by Lee in 2023. In particular, together with results from…
Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering…
The classical Kruskal-Katona theorem gives a tight upper bound for the size of an $r$-uniform hypergraph $\mathcal{H}$ as a function of the size of its shadow. Its stability version was obtained by Keevash who proved that if the size of…
The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalize this result in two ways. First we define the expander property of 3-uniform hypergraphs and…
We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).
We prove that every $n$ vertex linear triple system with $m$ edges has at least $m^6/n^7$ copies of a pentagon, provided $m>100 \, n^{3/2}$. This provides the first nontrivial bound for a question posed by Jiang and Yepremyan. More…
Every Steiner triple system is a uniform hypergraph. The coloring of hypergraph and its special case Steiner triple systems, {STS}$(v)$, is studied extensively. But the defining set of the coloring of hypergraph even its special case…
We introduce a uniform method of proof for the following results. For {\em each} of the following conditions, there are $2^{\aleph_0}$ families of Steiner systems, satisfying that condition: i) Theorem~2.2.4: (extending \cite{Chicoetal})…
A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erd\H{o}s conjectured that one can find so-called `sparse' Steiner triple systems. Roughly speaking, the…
A system of three particles undergoing inelastic collisions in arbitrary spatial dimensions is studied with the aim of establishing the domain of ``inelastic collapse''---an infinite number of collisions which take place in a finite time.…
A pair $(\mathcal{A},\mathcal{B})$ of families of subsets of an $n$-element set is called cancellative if whenever $A,A'\in\mathcal{A}$ and $B\in\mathcal{B}$ satisfy $A\cup B=A'\cup B$, then $A=A'$, and whenever $A\in\mathcal{A}$ and…