The size-Ramsey number of a graph F is the smallest number of edges in a graph G with the Ramsey property for F, that is, with the property that any 2-colouring of the edges of G contains a monochromatic copy of F. We prove that the size-Ramsey number of the grid graph on n × n vertices is bounded from above by n 3+o(1).

Estimating the potential impacts of climate change requires understanding the ability of agents to adapt to changes in their climate. This paper uses panel data from India spanning from 1956 to 1999 to investigate the ability of farmers to adapt. To identify adaptation, the author exploits persistent, multidecadal monsoon regimes during which droughts or floods are more common. These regimes generate medium-run variation in average rainfall, and there is spatial variation in the timing of the regimes. Using a fixed-effects strategy, she tests whether farmers have adapted to the medium-run rainfall variation induced by the monsoon regimes. The author finds evidence that farmers adjust their irrigation investments and their crop portfolios in response to the medium-run rainfall variation. However, adaptation only recovers a small fraction of the profits farmers have lost due to adverse climate variation.

Let Δ ≥ 2 be a fixed integer. We show that the random graph ${\mathcal{G}_{n,p}}$ with $p\gg (\log n/n)^{1/\Delta}$ is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Δ and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges from ${\mathcal{G}_{n,p}}$ in such a way that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all such H.

For c ∈ (0,1) let n(c) denote the set of n-vertex perfect graphs with density c and let n(c) denote the set of n-vertex graphs without induced C5 and with density c.

We show that with otherwise, where H is the binary entropy function.

Further, we use this result to deduce that almost all graphs in n(c) have homogeneous sets of linear size. This answers a question raised by Loebl and co-workers.

In 1978, Dhar suggested a model of a lattice gas whose states are partial orders. In this context he raised the question of determining the number of partial orders with a fixed number of comparable pairs. Dhar conjectured that in order to find a good approximation to this number, it should suffice to enumerate families of layer posets. In this paper we prove this conjecture and thereby prepare the ground for a complete answer to the question.

We provide sufficient conditions for packing two hypergraphs. The emphasis is on the asymptotic case when one of the hypergraphs has a bounded degree and the other is dense. As an application, we give an alternative proof for the bipartite case of the recently developed Blow-up Lemma [12].