The size‐Ramsey number of short subdivisions

Abstract

The r-size-Ramsey numberR r (H) of a graph H is the smallest number of edges a graph G can have such that for every edge-coloring of G with r colors there exists a monochromatic copy of H in G. For a graph H, we denote by H q the graph obtained from H by subdividing its edges with q − 1 vertices each. In a recent paper of Kohayakawa, Retter and Rödl, it is shown that for all constant integers q, r ≥ 2 and every graph H on n vertices and of bounded maximum degree, the r-size-Ramsey number of H q is at most (log n) 20(q−1) n 1+1∕q , for n large enough. We improve upon this result using a significantly shorter argument by showing thatR r (H q ) ≤ O(n 1+1∕q ) for any such graph H.

Divisible subdivisions

“…• There are several ways to improve the bound for f (H, q) in the proof of Theorem 1. In particular, one can use the constructions of Ramsey graphs for subdivisions given in [8,5]. In addition, it is possible to take k = ∆(Γ), where Γ is the Ramsey graph used, and M = 2|E(Γ)| + f 1 (q, k).…”

The size‐Ramsey number of cubic graphs

“…As pnp 2 q " np by the assumption on p, one can expect that the -th neighbourhood of each vertex v P N 1 v 1 contains almost all vertices in both N 1 v `1 and N 1 v t´ `1 . A minor modification of [12,Corollary 2.5] gives precisely this. Namely, it shows that there exists a vertex v P N 1 v 1 such that, for all but ε|N 1 v `1 | vertices w P N 1 v `1 , there exists a path from v to w (in G) with one vertex in each of N 1 v 2 , .…”

“…As $$(np^2)^\ell \gg np$$ by the assumption on $$p$$, one can expect that the $$\ell$$th neighbourhood of each vertex $$v \in N_{v_1}^{\prime }$$ contains almost all vertices in both $$N_{v_{\ell +1}}^{\prime }$$ and $$N_{v_{t-\ell +1}}^{\prime }$$. A minor modification of [13, Corollary 2.5] gives precisely this. Namely, it shows that there exists a vertex $$v \in N_{v_1}^{\prime }$$ such that, for all but $$\varepsilon |N_{v_{\ell +1}}^{\prime }|$$ vertices $$w \in N_{v_{\ell +1}}^{\prime }$$, there exists a path from $$v$$ to $$w$$ (in $$G$$) with one vertex in each of $$N_{v_2}^{\prime }, \ldots\,, N_{v_\ell }^{\prime }$$ and similarly for $$N_{v_{t-\ell +1}...$…”

Divisible subdivisions

“… There are several ways to improve the bound for in the proof of Theorem 1. In particular, one can use the constructions of Ramsey graphs for subdivisions given in [5,9] as a seed for our proof. In addition, it is possible to take , where is the Ramsey graph used, and .…”