We describe recent advances in the study of random analogues of combinatorial theorems.
The Ramsey number r k ( s , n ) r_k(s,n) is the minimum N N such that every red-blue coloring of the k k -tuples of an N N -element set contains a red set of size s s or a blue set of size n n , where a set is called red (blue) if all k k -tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r k ( s , n ) r_k(s,n) for k ≥ 3 k \geq 3 and s s fixed. In particular, we show that \[ r 3 ( s , n ) ≤ 2 n s − 2 log n , r_3(s,n) \leq 2^{n^{s-2}\log n}, \] which improves by a factor of n s − 2 / polylog n n^{s-2}/\textrm {polylog}\,n the exponent of the previous upper bound of Erdős and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there is a constant c > 0 c>0 such that \[ r 3 ( s , n ) ≥ 2 c s n log ( n s + 1 ) r_3(s,n) \geq 2^{c \, sn \, \log (\frac {n}{s}+1)} \] for all 4 ≤ s ≤ n 4 \leq s \leq n . For constant s s , this gives the first superexponential lower bound for r 3 ( s , n ) r_3(s,n) , answering an open question posed by Erdős and Hajnal in 1972. Next, we consider the 3 3 -color Ramsey number r 3 ( n , n , n ) r_3(n,n,n) , which is the minimum N N such that every 3 3 -coloring of the triples of an N N -element set contains a monochromatic set of size n n . Improving another old result of Erdős and Hajnal, we show that \[ r 3 ( n , n , n ) ≥ 2 n c log n . r_3(n,n,n) \geq 2^{n^{c \log n}}. \] Finally, we make some progress on related hypergraph Ramsey-type problems.
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress.
We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant C such that
Given a labeled graph H with vertex set {1, 2,. .. , n}, the ordered Ramsey number r < (H) is the minimum N such that every two-coloring of the edges of the complete graph on {1, 2,. .. , N } contains a copy of H with vertices appearing in the same order as in H. The ordered Ramsey number of a labeled graph H is at least the Ramsey number r(H) and the two coincide for complete graphs. However, we prove that even for matchings there are labelings where the ordered Ramsey number is superpolynomial in the number of vertices. Among other results, we also prove a general upper bound on ordered Ramsey numbers which implies that there exists a constant c such that r < (H) ≤ r(H) c log 2 n for any labeled graph H on vertex set {1, 2,. .. , n}.
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