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In this paper, we study an analogue of size-Ramsey numbers for vertex colorings. For a given number of colors r and a graph G the vertex size-Ramsey number of G, denoted by Rˆv(G,r), is the least number of edges in a graph H with the property that any r-coloring of the vertices of H yields a monochromatic copy of G. We observe that Ωr(Δn)=Rˆv(G,r)=Or(n2) for any G of order n and maximum degree ...
The well‐known Ramsey number is the smallest integer n such that every ‐free graph of order n contains an independent set of size u. In other words, it contains a subset of u vertices with no K2. Erdős and Rogers introduced a more general problem replacing K2 by for . Extending the problem of determining Ramsey numbers they defined the numbers
where the minimum is taken over all...
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