Ordered graphs and large bi-cliques in intersection graphs of curves

@article{Pach2019OrderedGA,
  title={Ordered graphs and large bi-cliques in intersection graphs of curves},
  author={J. Pach and Istv{\'a}n Tomon},
  journal={Eur. J. Comb.},
  year={2019},
  volume={82}
}
An ordered graph $G_<$ is a graph with a total ordering $<$ on its vertex set. A monotone path of length $k$ is a sequence of vertices $v_1<v_2<\ldots<v_k$ such that $v_iv_{j}$ is an edge of $G_<$ if and only if $|j-i|=1$. A bi-clique of size $m$ is a complete bipartite graph whose vertex classes are of size $m$. We prove that for every positive integer $k$, there exists a constant $c_k>0$ such that every ordered graph on $n$ vertices that does not contain a monotone path of length $k$ as an… Expand
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References

SHOWING 1-10 OF 24 REFERENCES
On the Chromatic Number of Disjointness Graphs of Curves
TLDR
The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist K_k-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least $\Omega(k^{4})$ colors. Expand
Separators in region intersection graphs
TLDR
The preceding result implies that every string graph with m edges has a balanced separator of size $O(\sqrt{m})$. Expand
Trees and linear anticomplete pairs
We prove a conjecture of Liebenau, Pilipczuk, and the last two authors, that for every forest $H$ there exists $\epsilon>0$, such that if $G$ has $n\ge 2$ vertices and does not contain $H$ as anExpand
Almost all string graphs are intersection graphs of plane convex sets
TLDR
It is obtained that {\em almost all} string graphs on $n$ vertices are intersection graphs of plane convex sets. Expand
A Bipartite Analogue of Dilworth’s Theorem
  • J. Fox
  • Mathematics, Computer Science
  • Order
  • 2006
Let m(n) be the maximum integer such that every partially ordered set P with n elements contains two disjoint subsets A and B, each with cardinality m(n), such that either every element of A isExpand
Turán-type results for partial orders and intersection graphs of convex sets
We prove Ramsey-type results for intersection graphs of geometric objects. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0Expand
Near-Optimal Separators in String Graphs
  • J. Matousek
  • Mathematics, Computer Science
  • Combinatorics, Probability and Computing
  • 2013
TLDR
Fox and Pach proved that G has a separator consisting of $O(m^{3/4}\sqrt{\log m})$ vertices, and they conjectured that the bound of O(\sqrt m)$ actually holds. Expand
Turán-Type Results for Complete h-Partite Graphs in Comparability and Incomparability Graphs
TLDR
It is shown that if the edge density of G(P, <) is strictly larger than 1 − 1/(2h − 2)r, then P contains h disjoint sets A1, ... , Ah such that A1 <j... <jAh holds for some 1 ≤ j ≤ r, and |A1| = ... = |Ah| = Ω(|P|). Expand
A Ramsey-Type Theorem for Orderings of a Graph
TLDR
It is shown that for any graph G on n vertices, there is a number N (of order at most $n^3 (\log n)^2 $) and a graph H on N vertices such that there is an order-isomorphism from G into H. Expand
Supersaturated graphs and hypergraphs
We shall consider graphs (hypergraphs) without loops and multiple edges. Let ℒ be a family of so called prohibited graphs and ex (n, ℒ) denote the maximum number of edges (hyperedges) a graphExpand
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