Results 1 
7 of
7
Sharp Bounds on DavenportSchinzel Sequences of Every Order
, 2013
"... One of the oldest unresolved problems in extremal combinatorics is to determine the maximum length of DavenportSchinzel sequences, where an orders DS sequence is defined to be one over an nletter alphabet that avoids alternating subsequences of the form a · · · b · · · a · · · b · · · wit ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
One of the oldest unresolved problems in extremal combinatorics is to determine the maximum length of DavenportSchinzel sequences, where an orders DS sequence is defined to be one over an nletter alphabet that avoids alternating subsequences of the form a · · · b · · · a · · · b · · · with length s + 2. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since become an indispensable tool in computational geometry and the analysis of discrete geometric structures. Let λs(n) be the extremal function for such sequences. What is λs asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch) when s is even or s ≤ 3. However, since the work of Agarwal, Sharir, and Shor in the 1980s there has been a persistent gap in our understanding of the odd orders, a gap that is just as much qualitative as quantitative. In this paper we establish the following bounds on λs(n) for every order s. n s = 1 2n − 1 s = 2 ⎪ ⎨ 2nα(n) + O(n) s = 3 λs(n) = Θ(n2 α(n) ) s = 4 Θ(nα(n)2 α(n) ) s = 5 n2 (1+o(1))αt (n)/t! s ≥ 6, t = ⌊ s−2 2 ⌋ These results refute a conjecture of Alon, Kaplan, Nivasch, Sharir, and Smorodinsky and run counter to common sense. When s is odd, λs behaves essentially like λs−1.
An Improved Bound on the Number of Unit Area Triangles
, 2009
"... We show that the number of unitarea triangles determined by a set of n points in the plane is O(n 9/4+ε), for any ε> 0, improving the recent bound O(n 44/19) of Dumitrescu et al. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We show that the number of unitarea triangles determined by a set of n points in the plane is O(n 9/4+ε), for any ε> 0, improving the recent bound O(n 44/19) of Dumitrescu et al.
Applications of the Canonical Ramsey Theorem to Geometry
, 2013
"... Let {p1,..., pn} ⊆ R d. We think of d ≤ n. How big is the largest subset X of points such that all of the distances determined by elements of () X 2 are different? We show that X is at least Ω((n 1/(6d) (log n) 1/3)/d 1/3). This is not the best known; however the technique is new. Assume that no th ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Let {p1,..., pn} ⊆ R d. We think of d ≤ n. How big is the largest subset X of points such that all of the distances determined by elements of () X 2 are different? We show that X is at least Ω((n 1/(6d) (log n) 1/3)/d 1/3). This is not the best known; however the technique is new. Assume that no three of the original points are collinear. How big is the largest subset X of points such that all of the areas determined by elements of () X 3 are different? We show that, if d = 2 then X is at least Ω((log log n) 1/186), and if d = 3 then X is at least Ω((log log n) 1/396). We also obtain results for countable sets of points in R d. All of our results use variants of the canonical Ramsey theorem and some geometric lemmas. 1
A Higher Dimensional Version of a Problem of Erdős
, 2013
"... Let {p1,..., pn} ⊆ Rd. We think of d ≪ n. How big is the largest subset X of points such that all of the distances determined by elements of () X are different? We ..."
Abstract
 Add to MetaCart
(Show Context)
Let {p1,..., pn} ⊆ Rd. We think of d ≪ n. How big is the largest subset X of points such that all of the distances determined by elements of () X are different? We
Large Subsets of Points with all Pairs (Triples) Having Different Distances (Areas)
, 2013
"... Let {p1,..., pn} ⊆ R d. We think of d ≤ n. How big is the largest subset X of points such that all of the distances determined by elements of () X 2 are different? We show that X is at least Ω((n1/(6d) (log n) 1/3)/d1/3). Assume that no three of the original points are collinear. How big is the lar ..."
Abstract
 Add to MetaCart
(Show Context)
Let {p1,..., pn} ⊆ R d. We think of d ≤ n. How big is the largest subset X of points such that all of the distances determined by elements of () X 2 are different? We show that X is at least Ω((n1/(6d) (log n) 1/3)/d1/3). Assume that no three of the original points are collinear. How big is the largest subset X of points such that all of the areas determined by elements of () X 3 are different? We show that, if d = 2 then X is at least Ω((log log n) 1/2901), and if d = 3 then X is at least Ω((log log n) 1/27804). All of our results use variants of the canonical Ramsey theorem and some geometric lemmas. 1