We study the problem of finding the largest numberT(n,m) of ternary vectors of lengthnsuch that for any three distinct vectors there are at leastmcoordinates where they pairwise differ. This problem is a special case of the perfectk-hashing problem in theoretical computer science, corresponding to thek= 3 case. Form= 1, we get the classical trifference problem which is wide open. We prove upper and lower bounds onT(n,m) for various ranges of the parametermand determine the phase transition threshold onm=m(n) whereT(n,m) jumps from constant to exponential in n. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters. Moreover, we relate the trifference problem to the sunflower conjecture.

We study exak(n,F), the largest number of edges in an n-vertex graph that contains exactly k copies of a given subgraph F. The case k=0 is the Turán number ex(n,F) that is among the most studied parameters in extremal graph theory. We show that for any F and k, exak(n,F)=(1+o(1))ex(n,F) and determine the exact values of exak(n,K3) and exa1(n,Kr) for n large enough. We also explore a connection to the following well-known problem in search theory. We are given a graph of order n that consists of an unknown copy of F and some isolated vertices. We can ask pairs of vertices as queries, and the answer tells us whether there is an edge between those vertices. Our goal is to describe the graph using as few queries as possible. Aigner and Triesch in 1990 showed that the number of queries needed is at least n2−exa1(n,F). Among other results we show that the number of queries that were answered NO is at least n2−exa1(n,F).

For an integer $k \geq 2$, an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $<$ of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal{H},\mathcal{G})$ of two ordered $k$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{G}$ is the smallest $N \in \mathbb{N}$ such that every red-blue coloring of the hyperedges of the ordered complete $k$-uniform hypergraph $\mathcal{K}^{(k)}_N$ on $N$ vertices contains a blue copy of $\mathcal{H}$ or a red copy of $\mathcal{G}$. The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs. In particular, we prove that for all $d,n \in \mathbb{N}$ and for every ordered 3-uniform hypergraph $\mathcal{H}$ on $n$ vertices with maximum degree $d$ and with interval chromatic number 3 there is an $\varepsilon=\varepsilon(d)>0$ such that $\overline{R}(\mathcal{H},\mathcal{H}) \leq 2^{O(n^{2-\varepsilon})}.$ In fact, we prove this upper bound for the number $\overline{R}(\mathcal{G},\mathcal{K}^{(3)}_3(n))$, where $\mathcal{G}$ is an ordered 3-uniform hypergraph with $n$ vertices and maximum degree $d$, and $\mathcal{K}^{(3)}_3(n)$ is the ordered complete tripartite hypergraph with consecutive color classes of size $n$. We show that this bound is not far from the truth by proving $\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(n)) \geq 2^{\Omega(n\log{n})}$ for some fixed ordered 3-uniform hypergraph $\mathcal{H}$.

Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m(G). It was shown by Saks and Werman that m(Kn)=n−b(n), where b(n) is the number of 1’s in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n−b(n)≤m(G)≤n−1. We show that for any tree T on an even number of vertices we have m(T)=n−1, and that for any tree T on an odd number of vertices, we have n−65≤m(T)≤n−2. Our proof uses results about the weighted version of the problem for Kn, which may be of independent interest. We also exhibit a sequence Gn of graphs with m(Gn)=n−b(n) such that Gn has O(nb(n)) edges and n vertices.

For a graph G, a hypergraph $\mathcal{H}$ is a Berge copy of G (or a Berge-G in short) if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We denote the family of r-uniform hypergraphs that are Berge copies of G by $B^rG$. For families of r-uniform hypergraphs $\mathbf{H}$ and $\mathbf{H}'$, we denote by $R(\mathbf{H},\mathbf{H}')$ the smallest number n such that in any red-blue coloring of the (hyper)edges of $\mathcal{K}_n^r$ (the complete r-uniform hypergraph on n vertices) there is a monochromatic blue copy of a hypergraph in $\mathbf{H}$ or a monochromatic red copy of a hypergraph in $\mathbf{H}'$. $R^c(\mathbf{H})$ denotes the smallest number n such that in any coloring of the hyperedges of $\mathcal{K}_n^r$ with c colors, there is a monochromatic copy of a hypergraph in $\mathbf{H}$. In this paper we initiate the general study of the Ramsey problem for Berge hypergraphs, and show that if $r> 2c$, then $R^c(B^rK_n)=n$. In the case r = 2c, we show that $R^c(B^rK_n)=n+1$, and if G is a noncomplete graph on n vertices, then $R^c(B^rG)=n$, assuming n is large enough. In the case $r < 2c$ we also obtain bounds on $R^c(B^rK_n)$. Moreover, we also determine the exact value of $R(B^3T_1,B^3T_2)$ for every pair of trees T_1 and T_2.

In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph ${\mathcal H}$ is a $t$-heavy copy of a graph $F$ if there is a copy of $F$ on its vertex set such that each edge of $F$ is contained in at least $t$ hyperedges of ${\mathcal H}$. ${\mathcal H}$ is a $t$-wise Berge copy of $F$ if additionally for distinct edges of $F$ those $t$ hyperedges are distinct. We extend known upper bounds on the Turán number of Berge hypergraphs to the $t$-wise Berge hypergraphs case. We asymptotically determine the Turán number of $t$-heavy and $t$-wise Berge copies of long paths and cycles and exactly determine the Turán number of $t$-heavy and $t$-wise Berge copies of cliques. In the case of 3-uniform hypergraphs, we consider the problem in more details and obtain additional results.

We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal nonpercolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.

We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set of points in the plane $${\mathcal {S}} \subset {\mathbb {R}}^2$$ can be 2-colored such that every axis-parallel square that contains at least m points from $${\mathcal {S}}$$ contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering 2-coloring points with respect to homothets of a fixed parallelogram.

We prove that if an $n$-vertex graph $G$ can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then $G$ has $O(n)$ edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal $1$-planar and fan-planar graphs.