The size Ramsey number of short subdivisions of bounded degree graphs

5 For graphs G and F , write G → (F )` if any coloring of the edges of G with ` colors yields 6 a monochromatic copy of the graph F . Let positive integers h and d be given. Suppose S 7 is obtained from a graph S with s vertices and maximum degree d by subdividing its edges h 8 times (that is, by replacing the edges of S by paths of length h+ 1). We prove that there exits 9 a graph G with no more than (log s)s edges for which G→ (S)` holds, provided 10 that s ≥ s0(h, d, `), where s0(h, d, `) is some constant that depends only on h, d, and `. We 11 also extend this result to the case in which Q is a graph with maximum degree d on q vertices 12 with the property that every pair of vertices of degree greater than 2 are distance at least h+ 1 13 apart. This complements work of Pak regarding the size Ramsey number of ‘long subdivisions’ 14 of bounded degree graphs. 15

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