Short proofs of some extremal results III

@article{Conlon2020ShortPO,
  title={Short proofs of some extremal results III},
  author={David Conlon and Jacob Fox and Benny Sudakov},
  journal={Random Structures \& Algorithms},
  year={2020},
  volume={57},
  pages={958 - 982}
}
We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short. 

Ramsey numbers with prescribed rate of growth

. Let R ( G ) be the two-colour Ramsey number of a graph G . In this note, we prove that for any non-decreasing function n 6 f ( n ) 6 R ( K n ), there exists a sequence of connected graphs ( G n ) n

Ramsey numbers upon vertex deletion

Given a graph G , its Ramsey number r ( G ) is the minimum N so that every two-coloring of E ( K N ) contains a monochromatic copy of G . It was conjectured by Conlon, Fox, and Sudakov that if one

Fan-complete Ramsey numbers

We consider Ramsey numbers r ( G, H ) with tight lower bounds, namely, where χ ( G ) denotes the chromatic number of G and | H | denotes the number of vertices in H . We say H is G -good if the

Ramsey non-goodness involving books

In 1983, Burr and Erd˝os initiated the study of Ramsey goodness problems. Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erd˝os, in which the bounds on the

Infinite Sperner's theorem

Ramsey goodness of books revisited

The Ramsey number r ( G, H ) is the minimum N such that every graph on N vertices contains G as a subgraph or its complement contains H as a subgraph. For integers n ≥ k ≥ 1, the k -book B k,n is the

Ramsey simplicity of random graphs

A graph G is q-Ramsey for another graph H if in any q-edge-colouring of G there is a monochromatic copy of H, and the classic Ramsey problem asks for the minimum number of vertices in such a graph.

References

SHOWING 1-10 OF 65 REFERENCES

Connected matchings and Hadwiger's conjecture

TLDR
Hadwiger's conjecture implies that any graph G on n vertices contains a $K_{\lceil {n\over \alpha(G)}\rceil}$ as a minor.

On a Ramsey type theorem

A Note on Ramsey Numbers

On the number of complete subgraphs contained in certain graphs

Books Versus triangles

TLDR
This short proof uses the triangle removal lemma, although there is another approach which avoids this.

On the number of complete subgraphs and circuits contained in graphs

and denote by KP the complete graph of p vertices . A well known theorem of TUR.áN [6] states that every 9(n ; m(n, p) + 1) contains a Kp and that this result is best possible. Thus in particular

Books versus Triangles at the Extremal Density

TLDR
This work makes a conjecture about this dependency and proves this conjecture for $\beta = 1/6$ and for $0.2495 \leq \beta < 1/4$, answering Mubayi's question in these ranges.

Complete Minors and Independence Number

  • J. Fox
  • Mathematics
    SIAM J. Discret. Math.
  • 2010
TLDR
This work shows that G, a graph with n vertices and independence number $\alpha", contains a clique minor of order larger than $504n/\alpha$.

An Infinite Sidon Sequence

Abstract We show the existence of an infinite Sidon sequence such that the number of elements in [1,  N ] is N 2 −1+ o (1) for all large N .

On an extremal problem in graph theory

G ( n;l ) will denote a graph of n vertices and l edges. Let f 0 ( n, k ) be the smallest integer such that there is a G ( n;f 0 (n, k )) in which for every set of k vertices there is a vertex joined
...