Almost all triple systems with independent neighborhoods are semi-bipartite

The neighborhood of a pair of vertices $u,v$ in a triple system is the set of vertices $w$ such that $uvw$ is an edge. A triple system \HH is semi-bipartite if its vertex set contains a vertex subset $X$ such that every edge of \HH intersects $X$ in exactly two points. It is easy to see that if \HH is semi-bipartite, then the neighborhood of every pair of vertices in \HH is an independent set. We show a partial converse of this statement by proving that almost all triple systems with vertex sets $[n]$ and independent neighborhoods are semi-bipartite. Our result can be viewed as an extension of the Erd\H os-Kleitman-Rothschild theorem to triple systems. The proof uses the Frankl-R\"odl hypergraph regularity lemma, and stability theorems. Similar results have recently been proved for hypergraphs with various other local constraints

Towards the map of quantum gravity

In this paper we point out some possible links between different approaches to quantum gravity and theories of the Planck scale physics. In particular, connections between Loop Quantum Gravity, Causal Dynamical Triangulations, Ho\v{r}ava-Lifshitz gravity, Asymptotic Safety scenario, Quantum Graphity, deformations of relativistic symmetries and nonlinear phase space models are discussed. The main focus is on quantum deformations of the Hypersurface Deformations Algebra and Poincar\'{e} algebra, nonlinear structure of phase space, the running dimension of spacetime and nontrivial phase diagram of quantum gravity. We present an attempt to arrange the observed relations in the form of a graph, highlighting different aspects of quantum gravity. The analysis is performed in the spirit of a mind map, which represents the architectural approach to the studied theory, being a natural way to describe the properties of a complex system. We hope that the constructed graphs (maps) will turn out to be helpful in uncovering the global picture of quantum gravity as a particular complex system and serve as a useful guide for the researchers.Comment: 27 pages, 7 figures, v2 style and presentation significantly improved, additional remarks and references added, v3 two figures modified, several important clarifications and references adde

Turan Problems and Shadows III: expansions of graphs

The expansion $G^+$ of a graph $G$ is the $3$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a new vertex disjoint from $V(G)$ such that distinct edges are enlarged by distinct vertices. Let $ex_3(n,F)$ denote the maximum number of edges in a $3$-uniform hypergraph with $n$ vertices not containing any copy of a $3$-uniform hypergraph $F$. The study of $ex_3(n,G^+)$ includes some well-researched problems, including the case that $F$ consists of $k$ disjoint edges, $G$ is a triangle, $G$ is a path or cycle, and $G$ is a tree. In this paper we initiate a broader study of the behavior of $ex_3(n,G^+)$. Specifically, we show $$ex_3(n,K_{s,t}^+) = \Theta(n^{3 - 3/s})$$ whenever $t > (s - 1)!$ and $s \geq 3$. One of the main open problems is to determine for which graphs $G$ the quantity $ex_3(n,G^+)$ is quadratic in $n$. We show that this occurs when $G$ is any bipartite graph with Tur\'{a}n number $o(n^{\varphi})$ where $\varphi = \frac{1 + \sqrt{5}}{2}$, and in particular, this shows $ex_3(n,Q^+) = \Theta(n^2)$ where $Q$ is the three-dimensional cube graph