A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen…

Let $U(N)$ denote the maximal length of arithmetic progressions in a random uniform subset of $\{0,1\}^N$. By an application of the Chen-Stein method, we show that $U(N)- 2 \log(N)/\log(2)$ converges in law to an extreme type (asymmetric) distribution. The same…