Ramsey goodness of trees in random graphs

Abstract

For graphs G,H$$ G,H $$ and a family of graphs ℱ$$ \mathcal{F} $$, we write G→false(H,ℱfalse)$$ G\to \left(H,\mathcal{F}\right) $$ to denote that every blue‐red coloring of the edges of G$$ G $$ contains either a blue copy of H$$ H $$, or a red copy of each F∈ℱ$$ F\in \mathcal{F} $$. For integers n$$ n $$ and D$$ D $$, let 𝒯(n,D) denote the family of all trees with n$$ n $$ edges and maximum degree at most D$$ D $$. We prove that for each r,D⩾2$$ r,D\geqslant 2 $$, there exist constants C,C′>0$$ C,{C}^{\prime }>0 $$ such that if p⩾Cnprefix−2false/false(r+2false)$$ p\geqslant C{n}^{-2/\left(r+2\right)} $$ and N⩾rn+C′false/p$$ N\geqslant rn+{C}^{\prime }/p $$, then G(N,p)→(Kr+1,𝒯(n,D))with high probability. This is a random version of a well‐known result of Chvátal from 1977. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erdős–Sós conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.

Dirac-Type Conditions for Spanning Bounded-Degree Hypertrees

“…Suppose we are given an (n, d, λ)-graph G and a tree T which contains a set of leaves L of size |L| ≥ αn for some 0 < α ≪ 1/∆. In order to embed T , we will follow a similar approach as it has been done before for trees with many leaves (see [3,10,14,18] for instance). Roughly speaking, the idea is to first embed T − L and then find a matching between the image of the parents of L and the unoccupied vertices in G.…”

“…The main tool is a powerful embedding technique, sometimes called extendability methods or tree embeddings with rollbacks, which was first introduced by Friedman and Pippenger [7] in 1987 and subsequently improved by Haxell [11] in 2001. Here we will use a modern reformulation of this technique which is attributed to Glebov, Johannsen, and Krivelevich [8], and which has played a major role in the solution of several problems in the last few years (see [3,4,6,9,10,15,13,18,19] for instance). Roughly speaking, the extendability method (Lemma 3.13) says that if we are given a subgraph S i ⊂ G which is 'extendable' and G has good expansion properties, then we can extend S i by adding a leaf e i with one of its endpoints in V (S i ) and other in V (G) \ V (S i ) so that S i + e i remains extendable.…”