# A Strengthening of the Erd\H{o}s-Szekeres Theorem

@article{Balogh2020ASO, title={A Strengthening of the Erd\H\{o\}s-Szekeres Theorem}, author={J{\'o}zsef Balogh and Felix Christian Clemen and Emily Heath and Mikhail Lavrov}, journal={arXiv: Combinatorics}, year={2020} }

The Erdős-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$ edges or a blue copy of the monotone increasing path with $s$ edges. Although $rs+1$ is the minimum number of vertices needed for this result, not all edges of $K_{rs+1}$ are necessary. We prove the following surprising characterization of ordered graphs on $rs+1$ vertices with this coloring…

## References

SHOWING 1-10 OF 19 REFERENCES

### On some Multicolor Ramsey Properties of Random Graphs

- MathematicsSIAM J. Discret. Math.
- 2017

It is shown that $5n/2-15/2 \le \hat{R}(P_n) \le 74n$ for $n$ sufficiently large, which improves the previous lower bound and improves the upper bound.

### Erd\H{o}s-Szekeres theorem for multidimensional arrays

- Mathematics
- 2019

The classical Erdős-Szekeres theorem dating back almost a hundred years states that any sequence of $(n-1)^2+1$ distinct real numbers contains a monotone subsequence of length $n$. This theorem has…

### New Lower Bounds on the Size-Ramsey Number of a Path

- MathematicsElectron. J. Comb.
- 2022

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true…

### On-line Ramsey Numbers for Paths and Stars

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2008

On-line version of size-Ramsey numbers of graphs of graphs are studied via a game played between Builder and Painter to force Painter to create a monochromatic copy of a graph H in as few rounds as possible.

### On size Ramsey number of paths, trees, and circuits. I

- MathematicsJ. Graph Theory
- 1983

It is demonstrated that random graphs satisfy some interesting Ramsey type properties and are shown to be finite, simple and undirected graphs.

### Erdős–Szekeres theorem for multidimensional arrays

- MathematicsJournal of the European Mathematical Society
- 2022

The classical Erdős-Szekeres theorem dating back almost a hundred years states that any sequence of (n − 1) + 1 distinct real numbers contains a monotone subsequence of length n. This theorem has…

### Monochromatic paths in random tournaments

- MathematicsElectron. Notes Discret. Math.
- 2017

It is proved that, with high probability, any 2-edge colouring of a random tournament on n vertices contains a monochromatic path of length Ω ( n / log n ) which implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.