The size Ramsey number of short subdivisions of bounded degree graphs

For graphs G and F, write G→(F)ℓ if any coloring of the edges of G with ℓ colors yields a monochromatic copy of the graph F. Suppose S(h) is obtained from a graph S with s vertices and maximum degree d by subdividing its edges h times (that is, by replacing the edges of S by paths of length h + 1). We prove that there exists a graph G with no more than (log⁡s)20hs1+1/(h+1) edges for which G→(S(h))ℓ holds, provided that s≥s0(h,d,ℓ). We also extend this result to the case in which Q is a graph with maximum degree d on q vertices with the property that every pair of vertices of degree greater than 2 are distance at least h + 1 apart. This complements work of Pak regarding the size Ramsey number of “long subdivisions” of bounded degree graphs.