# Constructions in Ramsey theory

@article{Mubayi2018ConstructionsIR, title={Constructions in Ramsey theory}, author={Dhruv Mubayi and Andrew Suk}, journal={Journal of the London Mathematical Society}, year={2018}, volume={97} }

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4‐uniform Ramsey number r4(5,n) , and the same for the iterated (k−4) ‐fold logarithm of the k ‐uniform version rk(k+1,n) . This is the first improvement of the original exponential lower bound for r4(5,n) implicit in work of Erdős and Hajnal from 1972 and also improves the current best known bounds for larger k due to the authors. Second, we prove an upper bound for…

## 4 Citations

The Erdős–Hajnal hypergraph Ramsey problem

- Mathematics
- 2016

Given integers 2 ≤ t ≤ k+1 ≤ n, let gk(t, n) be the minimum N such that every red/blue coloring of the k-subsets of {1, . . . , N} yields either a (k + 1)-set containing t red k-subsets, or an n-set…

New lower bounds for hypergraph Ramsey numbers

- Mathematics
- 2017

The Ramsey number rk(s,n) is the minimum N such that for every red–blue coloring of the k ‐tuples of {1,…,N} , there are s integers such that every k ‐tuple among them is red, or n integers such that…

A Survey of Hypergraph Ramsey Problems

- MathematicsSpringer Optimization and Its Applications
- 2020

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple…

Independent sets in hypergraphs with a forbidden link

- Mathematics, Computer ScienceProceedings of the London Mathematical Society
- 2019

We prove that there exists a 3‐uniform hypergraph on N vertices with independence number O(logN/loglogN) in which there are at most two edges among any four vertices. This bound is tight and solves a…

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