On some Multicolor Ramsey Properties of Random Graphs

Abstract

The size-Ramsey numberR(F ) of a graph F is the smallest integer m such that there exists a graph G on m edges with the property that any colouring of the edges of G with two colours yields a monochromatic copy of F . In this paper, first we focus on the size-Ramsey number of a path P n on n vertices. In particular, we show that 5n/2 − 15/2 ≤R(P n ) ≤ 74n for n sufficiently large. (The upper bound uses expansion properties of random d-regular graphs.) This improves the previous lower bound,R(P n ) ≥ (1 + √ 2)n − O(1), due to Bollobás, and the upper bound,R(P n ) ≤ 91n, due to Letzter. Next we study long monochromatic paths in edge-coloured random graph G(n, p) with pn → ∞. Let α > 0 be an arbitrarily small constant. Recently, Letzter showed that a.a.s. any 2-edge colouring of G(n, p) yields a monochromatic path of length (2/3 − α)n, which is optimal. Extending this result, we show that a.a.s. any 3-edge colouring of G(n, p) yields a monochromatic path of length (1/2 − α)n, which is also optimal. In general, we prove that for r ≥ 4 a.a.s. any r-edge colouring of G(n, p) yields a monochromatic path of length (1/r − α)n. We also consider a related problem and show that for any r ≥ 2, a.a.s. any r-edge colouring of G(n, p) yields a monochromatic connected subgraph on (1/(r−1)−α)n vertices, which is also tight.

Long Cycles in Locally Expanding Graphs, with Applications

“…The next question to ask then is what is the hidden dependence on r in this linear in n bound. This question has recently been addressed by Dudek and Pra lat; they proved [20] thatR(P n , r) ≥ (r+3)r 4 n − O(r 2 ) for all large enough n. In Section 5 we will provide an alternative proof of the lower bound of the size Ramsey number of paths. As for the upper bound in terms of r, in the same paper it was shown thatR(P n , r) ≤ 33r4 r n for all large enough n. These two bounds are quite far apart in terms of dependence on r, leaving the exponential gap.…”

“…In the groundbreaking paper [4] Beck, resolving a $100 question of Erdős, proved thatR(P n , 2) = O(n), where P n is an n-vertex path. (Recently, there has been a race to improve the multiplicative constant in this bound, see [19,36,20]; the best result is due to Dudek and Pra lat, who showed in [20] thatR(P n , 2) ≤ 74n for large enough n.) Beck's argument is density-type and can easily be adapted to show thatR(P n , r) = O r (n). The next question to ask then is what is the hidden dependence on r in this linear in n bound.…”

Perfect matchings and Hamiltonian cycles in the preferential attachment model

“…(a) The following constants satisfy conditions (16)(17)(18) (b) The following constants also satisfy conditions (16)(17)(18) Similarly, for π 1 (G n 2m,m � ), we get:…”

“…Note that conditions (16)(17)(18)(19) can be trivially satisfied by taking any arbitrary x, y, z, d ∈ (0, 1) and w > 0 such that y < z, and then choosing m ∈ N sufficiently large and α > 0 sufficiently small. We leave the result with such flexibility as our goal will be to tune everything to get the constant m as small as possible.…”

Large monochromatic components in multicolored bipartite graphs

“…Also note that for the rest of the proof, if we have at most two monochromatic components covering all of V G ( ), then we are done since at least one of them has at least m n ( + )/2 vertices. Since x m > /3 1 and y n > /3 2 , every vertex in Y 2 has a red neighbor in X 1 and every vertex in X 1 has a red neighbor in Y 2 , so there are nontrivial red components R R , …, k 1 in G which cover all of ∪ X Y Suppose first that ℓ = 1. Of course we must have ≥ k 2 otherwise there would be at most two red components covering all of G and we are done.…”