A Positive Fraction Erdos - Szekeres Theorem

Abstract

We prove a fractional version of the Erdős-Szekeres theorem: for any k there is a constant c k > 0 such that any sufficiently large finite set X ⊂ R 2 contains k subsets Y 1 , . . . , Y k , each of size ≥ c k |X |, such that every set {y 1 , . . . , y k } with y i ∈ Y i is in convex position. The main tool is a lemma stating that any finite set X ⊂ R d contains "large" subsets Y 1 , . . . , Y k such that all sets {y 1 , . . . , y k } with y i ∈ Y i have the same geometric (order) type. We also prove several related results (e.g., the positive fraction Radon theorem, the positive fraction Tverberg theorem).

Homogeneous selections from hyperplanes

“…In Sections 3 and 4 we present two proofs for Lemma 1.4. The first proof, which provides a better estimate for the value of the constant b(d), uses duality and is based on a same type lemma for points, due to Bárány and Valtr [2] (see also [8]). The second proof is shorter, but it utilizes a far reaching generalization of the same type lemma to semi-algebraic relations of several variables, found by Fox, Gromov, Lafforgue, Naor, and Pach [5], see also Bukh and Hubard [4] for a quantitative form.…”

Problems and Results around the Erdös-Szekeres Convex Polygon Theorem

“…Maybe one can say more in the given geometric situation, for instance, the many convex position n-tuples come with some structure. The following theorem, due to Bárány and Valtr [7], shows that these n-tuples can be chosen homogeneously: We call this result the "homogeneous" Erdős-Szekeres theorem. The proof in [7] is based on another homogeneous statement, the so called same type lemma.…”

“…The following theorem, due to Bárány and Valtr [7], shows that these n-tuples can be chosen homogeneously: We call this result the "homogeneous" Erdős-Szekeres theorem. The proof in [7] is based on another homogeneous statement, the so called same type lemma. We state it in dimension d, but first a definition: Two n-tuples x 1 , .…”

“…The proof is based on the center-point theorem of Rado [7], or on Borsuk's theorem [37]. It uses a reformulation of the definition of same type: all transversals of Y 1 , .…”

Density and regularity theorems for semi-algebraic hypergraphs

“…, P k ) has same-type transversals if all of its transversals have the same order-type. Bárány and Valtr [9] showed that for d, k > 1, there exists a c = c(d, k) such that the following holds. Let P 1 , .…”