The size‐Ramsey number of powers of bounded degree trees

Abstract

Given a positive integer s, the s‐colour size‐Ramsey number of a graph H is the smallest integer m such that there exists a graph G with m edges with the property that, in any colouring of E(G) with s colours, there is a monochromatic copy of H. We prove that, for any positive integers k and s, the s‐colour size‐Ramsey number of the kth power of any n‐vertex bounded degree tree is linear in n. As a corollary, we obtain that the s‐colour size‐Ramsey number of n‐vertex graphs with bounded treewidth and bounded degree is linear in n, which answers a question raised by Kamčev, Liebenau, Wood and Yepremyan.

The size-Ramsey number of 3-uniform tight paths

“…Then we can assume by the pigeonhole principle w.l.o.g. (we will not use the triple u 1 u 2 w 1 ) that there are sets X j ⊆ X j with |X j | ≥ 2 for j ∈ [5] (here we use that |X j | ≥ 3) such that for all j ∈ [5] and every x ∈ X j the triple w 1 w 2 x is red. Now we can continue exactly as in the case where (4.2) holds, with u 1 u 2 replaced by w 1 w 2 throughout and extending the path with end-tuple (w 1 , w 2 ).…”

“…Our proof combines new ideas and the method developed by Clemens, Jenssen, Kohayakawa, Morrison, Mota, Reding, and Roberts [7] for estimating the size-Ramsey number of powers of paths (see also [5,12]). It is plausible that ideas from [12,5] may provide a strategy to solve the case with s ≥ 3 colours. However, the question whether the size-Ramsey number of a tight path is linear for hypergraphs with uniformity r ≥ 4 remains open and requires additional ideas.…”

Three early problems on size Ramsey numbers

“…Note that since the colors are complementary, if a pair is ε-regular in red, then it is also ε-regular in blue. 4 Call a part V i red if at least half its internal edges are red and blue otherwise. Without loss of generality, suppose that m ′ ≥ m/2 of the parts are red and reindex so that these red parts are V 1 , .…”

“…The fourth question, about the size Ramsey number of paths, was resolved by Beck [2], who proved the surprising result that r(P n ) = Θ(n) for the path P n with n vertices. This breakthrough inspired many of the subsequent developments in the field, such as the classic papers [3,17,20,23,28] and the more recent results in [4,5,6,12,13,14,18,19,22,25].…”

The size-Ramsey number of cubic graphs

“…In other words, when H " K n one cannot do better than taking G to be the smallest complete graph that is Ramsey for H. A significantly more interesting example is when H " P n , the path on n vertices, for which Beck [4] has shown that rpP n q " Opnq. Subsequent work has extended this result to many other familes of graphs, including boundeddegree trees [14], cycles [20], and, more recently, powers of paths and bounded-degree trees [5] and long subdivisions [11]. For some further recent developments, see [7,18,19,22,25].…”