Given graphs G and H and a positive integer q, say that G is q‐Ramsey for H, denoted G→(H)q, if every q‐coloring of the edges of G contains a monochromatic copy of H. The size‐Ramsey number truerˆ(H) of a graph H is defined to be truerˆ(H)=min{∣E(G)∣:G→(H)2}. Answering a question of Conlon, we prove that, for every fixed k, we have truerˆ(Pnk)=O(n), where Pnk is the kth power of the n‐vertex path Pn (ie, the graph with vertex set V(Pn) and all edges {u,v} such that the distance between u and v in Pn is at most k). Our proof is probabilistic, but can also be made constructive.

For an integer q ě 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H, yet no proper subgraph of G has this property then G is called q-Ramsey-minimal for H. Generalising a statement by Burr, Nešetřil and Rödl from 1977 we prove that, for q ě 3, if G is a graph that is not q-Ramsey for some graph H then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H, as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences.• For 2 ď r ă q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H.• For every q ě 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus, and chromatic number.• The collection tM q pHq : H is 3-connected or K 3 u forms an antichain, where M q pHq denotes the set of all graphs that are q-Ramsey-minimal for H.We also address the question which pairs of graphs satisfy M q pH 1 q " M q pH 2 q, in which case H 1 and H 2 are called q-equivalent. We show that two graphs H 1 and H 2 are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ě 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: Results by Nešetřil and Rödl and by Fox et al. imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours. *

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 n3+o(1).

Dados grafos G e H e um inteiro positivo q, dizemos que G é qRamsey para H se toda q-coloração das arestas de G contém uma c ópia monocromática de H. Denotamos essa propriedade por G Ñ pHqq. O nú mero de Ramsey relativo a arestas r^pHq de um grafo H é definido como r^pHq mint|EpGq| : G Ñ pHq2u. Respondendo uma pergunta sugerida por Conlon, provamos que r^pPnkq Opnq para todo k fixo, onde Pnk é a k-ésima potência do caminho com n vértices Pn, i.e., o grafo com conjunto de vértices V pPnq e todas as arestas tu; vu tais que a dist ância entre u e v em Pn é no máximo k.

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