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}
}
  • D. Mubayi, Andrew Suk
  • Published 22 November 2015
  • Mathematics
  • Journal of the London Mathematical Society
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… 
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References

SHOWING 1-10 OF 40 REFERENCES
Hypergraph Ramsey numbers
The Ramsey number rk(s, n) is the minimum N such that every red-blue coloring of the k-tuples of an N -element set contains a red set of size s or a blue set of size n, where a set is called red
Two-source dispersers for polylogarithmic entropy and improved ramsey graphs
  • Gil Cohen
  • Computer Science, Mathematics
    Electron. Colloquium Comput. Complex.
  • 2015
TLDR
This work significantly improves Erdős’ result and construct 2(loglogn)c-Ramsey graphs, for some universal constant c, and resolves the problem of explicitly constructing dispersers for two n-bit sources with entropy (n).
On Ramsey Like Theorems , Problems and Results
(1) n → (k1, . . . , k`)` means that if we split the r-tuples of a set, S, |S| = n into ` classes, than for some i, 1 ≤ i ≤ ` there is a subset Si ⊆ S, |Si| ≥ ki all of whose r-tuples are in the i-th
The triangle-free process and R(3,k)
The areas of Ramsey Theory and Random Graphs have been closely linked every since Erd\H{o}s' famous proof in 1947 that the 'diagonal' Ramsey numbers R(k) grow exponentially in k. In the early 1990s,
On generalized Ramsey numbers of Erdős and Rogers
2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction
TLDR
The main novelty comes in a bootstrap procedure which allows the Challenge-Response mechanism for detecting "entropy concentration" of [4] to be used with sources of less and less entropy, using recursive calls to itself.
2-source dispersers for $n^{o(1)}$ entropy, and Ramsey graphs beating the Frankl-Wilson construction
TLDR
An explicit disperser for two independent sources on n bits, each of min-entropy k = 2 1−α0 , for some small absolute constant α0 > 0 (N = 2 and K = 2), and an explicit N ×N Boolean matrix for which no K ×K sub-matrix is monochromatic.
The Ramsey Number R(3, t) Has Order of Magnitude t2/log t
TLDR
It is proved that R(3, t) is bounded below by (1 – o(1))t/2/log t times a positive constant, and it follows that R (3), the Ramsey number for positive integers s and t, has asymptotic order of magnitude t2/ log t.
Combinatorial Theorems on Classifications of Subsets of a Given Set
Given any positive integers k, n, ANT, there is a positive integer M which has the following property. If S = {1, 2, . . ., A1}, and A is any distribution of Q,(S) into k classes, there is always an
Ramsey-type Theorems with Forbidden Subgraphs
TLDR
This work answers the question in the affirmative that for every graph H, there exists an such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least and establishes several Ramsey type results for tournaments.
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