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Abstract We present a refinement of the classical alteration method for constructing ‐free graphs for suitable edge‐probabilities , we show that removing all edges in ‐copies of the binomial random graph does not significantly change the independence number. This differs from earlier alteration approaches of Erdős and Krivelevich, who obtained similar guarantees by removing one edge from each ‐copy (instead of all of them). We demonstrate the usefulness of our refined alternation method via two applications to online graph Ramsey games, where it enables easier analysis. 

Abstract The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Prałat noted that around  the clique chromatic number of the random graph  changes by  when we increase the edge‐probability  by , but left the details of this surprising “jump” phenomenon as an open problem. We settle this problem, that is, resolve the nature of this polynomial “jump” of the clique chromatic number of the random graph  around edge‐probability . Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of  up to logarithmic factors for any edge‐probability . 

Abstract We consider rooted subgraphs in random graphs, that is, extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question as to whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open. 

Given a fixed graph H, what is the (exponentially small) probability that the number X_Hof copies of Hin the binomial random graph G_n,pis at least twice its mean? Studied intensively since the mid 1990s, this so‐called infamous upper tail problem remains a challenging testbed for concentration inequalities. In 2011 DeMarco and Kahn formulated an intriguing conjecture about the exponential rate of decay of for fixed ε > 0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound. 

Abstract We introduce a non-increasing tree growth process $$((T_n,{\sigma}_n),\, n\ge 1)$$ , where T n is a rooted labelled tree on n vertices and σ n is a permutation of the vertex labels. The construction of ( T n , σ n ) from ( T n −1 , σ n −1 ) involves rewiring a random (possibly empty) subset of edges in T n −1 towards the newly added vertex; as a consequence T n −1 ⊄ T n with positive probability. The key feature of the process is that the shape of T n has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process. We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n , this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $$c\ln n$$ , c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed.