The $r$-blowup of a graph $F$, denoted by $F[r]$, is the graph obtained by replacing the vertices and edges of $F$ with independent sets of size $r$ and copies of $K_{r,r}$, respectively. For bipartite graphs $F$, very little is known about the order of magnitude of the Turán number of $F[r]$. In this paper we prove that ${ex}(n,C_6[2])=O(n^{5/3})$ and, more generally, for any positive integer $t$, ${ex}(n,\theta_{3,t}[2])=O(n^{5/3})$. This is tight when $t$ is sufficiently large.

The Erdos-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on$n$vertices is at most n. Erdös considered this to be one of his three most favorite combinatorial problems and offered a $500 reward for a proof of this conjecture. We prove this conjecture for every large n. Here, we also provide a randomised algorithm to find such a colouring in polynomial time with high probability.

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.