Random maximalH-free graphs

Ramsey theory constructions from hypergraph matchings

“…Eventually, by considering the random greedy independent set algorithm in hypergraphs, Bohman and Bennett [2] extended the lower bound to the hypergraph setting. Generalizing the results of Osthus and Taraz [20], upper bounds in the hypergraph setting were obtained by Kühn, Osthus and Taylor [19].…”

“…Bollobás and Riordan [10] obtained analogous bounds for F n (F) if F is a complete graph or cycle on four vertices. In 2001, Osthus and Taraz [20] generalized these results to all strictly 2-balanced graphs thus providing estimates for this large class of graphs that are tight up to logarithmic factors.…”

Ramsey theory constructions from hypergraph matchings

“…Eventually, by considering the random greedy independent set algorithm in hypergraphs, Bohman and Bennett [2] extended the lower bound to the hypergraph setting. Generalizing the results of Osthus and Taraz [20], upper bounds in the hypergraph setting were obtained by Kühn, Osthus and Taylor [19].…”

“…Bollobás and Riordan [10] obtained analogous bounds for F n (F) if F is a complete graph or cycle on four vertices. In 2001, Osthus and Taraz [20] generalized these results to all strictly 2-balanced graphs thus providing estimates for this large class of graphs that are tight up to logarithmic factors.…”

Prominent examples of flip processes

“…In particular, these results imply the currently best known lower bound for the off‐diagonal Ramsey number $$$ R\left(3,k\right) $$$. Among the numerous further papers studying the triangle‐free process and other $$$ H $$$‐free processes, let us highlight [4, 21, 28]. The last class of processes we mention before turning to the main subject of this paper are the Achlioptas processes , introduced in [1].…”

Minimizing the number of complete bipartite graphs in a Ks-saturated graph

“…Unfortunately, this random variable seems quite challenging to understand. If F satisfies a certain balancedness condition (that, e.g., complete graphs and cycles satisfy), and we let p = n − v(F )−2 e(F )−1 (log n) 1/(e(F )−1) , then many results on the F -free process show that the resulting graph behaves in certain ways like the Erdős-Rényi random graph with edge probability p. Bohman and Keevash [3] conjectured that the maximum degree of the final graph in the F -free process will be O(pn) and this was proven up to a logarithmic factor by Osthus and Taraz [19]. Bohman and Keevash also showed that in the early stages of the F -free process, the number of common neighbors of any fixed set of vertices is of the same order of magnitude as what it would be in a random graph with edge probability n − v(F )−2 e(F )−1 ([3] Corollary 1.5).…”

Ramsey theory constructions from hypergraph matchings

“…Eventually, by considering the random greedy independent set algorithm in hypergraphs, Bohman and Bennett [2] extended the lower bound to the hypergraph setting. Generalizing the results of Osthus and Taraz [20], upper bounds in the hypergraph setting were obtained by Kühn, Osthus and Taylor [19].…”

“…Bollobás and Riordan [10] obtained analogous bounds for F n (F) if F is a complete graph or cycle on four vertices. In 2001, Osthus and Taraz [20] generalized these results to all strictly 2-balanced graphs thus providing estimates for this large class of graphs that are tight up to logarithmic factors.…”

Prominent examples of flip processes

“…In particular, these results imply the currently best known lower bound for the off‐diagonal Ramsey number $$$ R\left(3,k\right) $$$. Among the numerous further papers studying the triangle‐free process and other $$$ H $$$‐free processes, let us highlight [4, 21, 28]. The last class of processes we mention before turning to the main subject of this paper are the Achlioptas processes , introduced in [1].…”

Minimizing the number of complete bipartite graphs in a Ks-saturated graph

“…Unfortunately, this random variable seems quite challenging to understand. If F satisfies a certain balancedness condition (that, e.g., complete graphs and cycles satisfy), and we let p = n − v(F )−2 e(F )−1 (log n) 1/(e(F )−1) , then many results on the F -free process show that the resulting graph behaves in certain ways like the Erdős-Rényi random graph with edge probability p. Bohman and Keevash [3] conjectured that the maximum degree of the final graph in the F -free process will be O(pn) and this was proven up to a logarithmic factor by Osthus and Taraz [19]. Bohman and Keevash also showed that in the early stages of the F -free process, the number of common neighbors of any fixed set of vertices is of the same order of magnitude as what it would be in a random graph with edge probability n − v(F )−2 e(F )−1 ([3] Corollary 1.5).…”

Ramsey theory constructions from hypergraph matchings

“…Eventually, by considering the random greedy independent set algorithm in hypergraphs, Bohman and Bennett [2] extended the lower bound to the hypergraph setting. Generalizing the results of Osthus and Taraz [20], upper bounds in the hypergraph setting were obtained by Kühn, Osthus and Taylor [19].…”

“…Bollobás and Riordan [10] obtained analogous bounds for F n (F) if F is a complete graph or cycle on four vertices. In 2001, Osthus and Taraz [20] generalized these results to all strictly 2-balanced graphs thus providing estimates for this large class of graphs that are tight up to logarithmic factors.…”

Prominent examples of flip processes

“…In particular, these results imply the currently best known lower bound for the off‐diagonal Ramsey number $$$ R\left(3,k\right) $$$. Among the numerous further papers studying the triangle‐free process and other $$$ H $$$‐free processes, let us highlight [4, 21, 28]. The last class of processes we mention before turning to the main subject of this paper are the Achlioptas processes , introduced in [1].…”

Minimizing the number of complete bipartite graphs in a Ks-saturated graph

“…Unfortunately, this random variable seems quite challenging to understand. If F satisfies a certain balancedness condition (that, e.g., complete graphs and cycles satisfy), and we let p = n − v(F )−2 e(F )−1 (log n) 1/(e(F )−1) , then many results on the F -free process show that the resulting graph behaves in certain ways like the Erdős-Rényi random graph with edge probability p. Bohman and Keevash [3] conjectured that the maximum degree of the final graph in the F -free process will be O(pn) and this was proven up to a logarithmic factor by Osthus and Taraz [19]. Bohman and Keevash also showed that in the early stages of the F -free process, the number of common neighbors of any fixed set of vertices is of the same order of magnitude as what it would be in a random graph with edge probability n − v(F )−2 e(F )−1 ([3] Corollary 1.5).…”