A positive fraction Erdos-Szekeres theorem and its applications

Abstract

A famous theorem of Erdos and Szekeres states that any sequence of $n$ distinct real numbers contains a monotone subsequence of length at least $\sqrt{n}$. Here, we prove a positive fraction version of this theorem. For $n > (k-1)^2$, any sequence $A$ of $n$ distinct real numbers contains a collection of subsets $A_1,\ldots, A_k \subset A$, appearing sequentially, all of size $s=Ω(n/k^2)$, such that every subsequence $(a_1,\ldots, a_k)$, with $a_i \in A_i$, is increasing, or every such subsequence is decreasing. The subsequence $S = (A_1,\ldots, A_k)$ described above is called block-monotone of depth $k$ and block-size $s$. Our theorem is asymptotically best possible and follows from a more general Ramsey-type result for monotone paths, which we find of independent interest. We also show that for any positive integer $k$, any finite sequence of distinct real numbers can be partitioned into $O(k^2\log k)$ block-monotone subsequences of depth at least $k$, upon deleting at most $(k-1)^2$ entries. We apply our results to mutually avoiding planar point sets and biarc diagrams in graph drawing.

Figures (6)

• Figure 1: (i) a $(3,2)$-configuration. (ii) a $(3,2,2)$-pattern.
• Figure 2: In proof of Lemma \ref{['partition_lemma_1']}, $S_{i'}\subset D(S_{i})\cap R(S_{i})$ for $i<i'$.
• Figure 3: Division of the plane into $9$ regions according to $(x_i,y_i),i=1,2$. Each ellipse represents a cluster of points as defined in the proof.
• Figure 4: The division of plane into regions according to $L,H,N$.
• Figure 5: An example when $A_i$'s are increasing. Each ellipse represents a cluster of points as defined in the proof.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Definition 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• proof : Proof of Theorem \ref{['path']}