Ramsey goodness of paths

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ the size of the smallest color class in a χ(H)- coloring of H. A graph G is called H-good if R(G, H) = (|G| − 1)(χ(H) − 1) + σ(H). The notion of Ramsey goodness was introduced by Burr and Erd˝os in 1983 and has been extensively studied since then. In this short note we prove that n-vertex path Pn is H-good for all n ≥ 4|H|. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan

Ramsey goodness of paths

Given a pair of graphs G and H, the Ramsey number R(G,H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If graph G is connected, it is well known and easy to show that R(G,H)≥(|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ the size of the smallest color class in a χ(H)-coloring of H. A graph G is called H-good if R(G,H)=(|G|−1)(χ(H)−1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then. In this short note we prove that n-vertex path Pn is H-good for all n≥4|H|. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan

Ramsey goodness of cycles

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known and easy to show that R(G, H) \geq (| G| - 1)(\chi (H) - 1) + \sigma (H), where \chi (H) is the chromatic number of H and \sigma (H) is the size of the smallest color class in a \chi (H)-coloring of H. A graph G is called H-good if R(G, H) = (| G| - 1)(\chi (H) - 1) + \sigma (H). The notion of Ramsey goodness was introduced by Burr and Erd\H os in 1983 and has been extensively studied since then. In this paper we show that if n \geq 1060| H| and \sigma (H) \geq \chi (H) 22, then the n-vertex cycle Cn is H-good. For graphs H with high \chi (H) and \sigma (H), this proves in a strong form a conjecture of Allen, Brightwell, and Skokan

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices