Let $\mathcal{F}$ be a family of $3$-uniform linear hypergraphs. The linear Turán number of $\mathcal F$ is the maximum possible number of edges in a $3$-uniform linear hypergraph on $n$ vertices which contains no member of $\mathcal{F}$ as a subhypergraph. In this paper we show that the linear Turán number of the five cycle $C_5$ (in the Berge sense) is $\frac{1}{3 \sqrt3}n^{3/2}$ asymptotically. We also show that the linear Turán number of the four cycle $C_4$ and $\{C_3, C_4\}$ are equal asmptotically, which is a strengthening of a theorem of Lazebnik and Verstraëte. We establish a connection between the linear Turán number of the linear cycle of length $2k+1$ and the extremal number of edges in a graph of girth more than $2k-2$. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang, we obtain that the linear Turán number of the linear cycle of length $2k+1$ is $\Theta(n^{1+\frac{1}{k}})$ for $k = 2, 3, 4, 6$.

Let $\mathcal{H}$ be a set of graphs. The planar Turán number, $ex_\mathcal{P}(n,\mathcal{H})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{H}$ as a subgraph. When $\mathcal{H}=\{H\}$ has only one element, we usually write $ex_\mathcal{P}(n,H)$ instead. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_5)$ and $ex_\mathcal{P}(n,K_4)$. Later on, we obtained sharper bound for $ex_\mathcal{P}(n,\{K_4,C_7\})$. In this paper, we give upper bounds of $ex_\mathcal{P}(n,\{K_4,C_5\})\leq {15\over 7}(n-2)$ and $ex_\mathcal{P}(n,\{K_4,C_6\})\leq {7\over 3}(n-2)$. We also give constructions which show the bounds are tight for infinitely many graphs.

Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particular, we show that if a set $P$ of $n$ points is in convex position, then the largest intersecting family of triangles with vertices in $P$ contains at most $(\frac{1}{4}+o(1))\binom{n}{3}$ triangles.

The motivation for extending secret sharing schemes to cases when either the set of players is infinite or the domain from which the secret and/or the shares are drawn is infinite or both, is similar to the case when switching to abstract probability spaces from classical combinatorial probability. It might shed new light on old problems, could connect seemingly unrelated problems, and unify diverse phenomena. Definitions equivalent in the finitary case could be very much different when switching to infinity, signifying their difference. The standard requirement that qualified subsets should be able to determine the secret has different interpretations in spite of the fact that, by assumption, all participants have infinite computing power. The requirement that unqualified subsets should have no, or limited information on the secret suggests that we also need some probability distribution. In the infinite case events with zero probability are not necessarily impossible, and we should decide whether bad events with zero probability are allowed or not. In this paper, rather than giving precise definitions, we enlist an abundance of hopefully interesting infinite secret sharing schemes. These schemes touch quite diverse areas of mathematics such as projective geometry, stochastic processes and Hilbert spaces. Nevertheless our main tools are from probability theory. The examples discussed here serve as foundation and illustration to the more theory oriented companion paper.

Since its formulation, Turán's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a $3$-uniform hypergraph $\mathcal{F}$ on $n$ vertices in which any five vertices span at least one edge, prove that $|\mathcal{F}| \ge (1/4 -o(1))\binom{n}{3}$. The construction showing that this bound would be best possible is simply $\binom{X}{3} \cup \binom{Y}{3}$ where $X$ and $Y$ evenly partition the vertex set. This construction has the following more general $(2p+1, p+1)$-property: any set of $2p+1$ vertices spans a complete sub-hypergraph on $p+1$ vertices. One of our main results says that, quite surprisingly, for all $p>2$ the $(2p+1,p+1)$-property implies the conjectured lower bound.

Let $p_0,\ldots,p_n$ be a finite sequence of points in an Euclidean space $\R^d$. Suppose that there is a (pointlike) particle sitting at each point $p_i$. In a ``legal'' move, any one of them can jump over another, landing on the other side, at exactly the same distance. Under what circumstances can we guarantee that for any $\varepsilon>0$ and any other sequence of points $q_0,\ldots, q_n\in\R^d$, there is a finite sequence of legal moves that takes the particle at $p_i$ to the $\varepsilon$-neighborhood of $q_i$, simultaneously for every $i$? We prove that this is possible if and only if the additive group generated by the vectors $p_1-p_0,\ldots,p_n-p_0$ is dense in $\R^d$.

A graph G = (V,E) is called fully regular if for every independent set $I\subset V$ , the number of vertices in $V\setminus$ I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph. As an application of our results, we give alternative proofs of two theorems of Stanley and Gao + Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph. As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.

Each member of an $n$-person team has a secret, say a password. The $k$ out of $n$ gruppen secret sharing requires that any group of $k$ members should be able to recover the secrets of the other $n-k$ members, while any group of $k-1$ or less members should have no information on the secret of other team member even if other secrets leak out. We prove that when all secrets are chosen independently and have size $s$, then each team member must have a share of size at least $(n-k)s$, and we present a scheme which achieves this bound when $s$ is large enough. This result shows a significant saving over $n$ independent applications of Shamir's $k$ out of $n-1$ threshold schemes which assigns shares of size $(n-1)s$ to each team member independently of $k$. We also show how to set up such a scheme without any trusted dealer, and how the secrets can be recovered, possibly multiple times, without leaking information. We also discuss how our scheme fits to the much-investigated multiple secret sharing methods.