Semi-algebraic Ramsey numbers

Given a finite point set $P \subset \mathbb{R}^d$, a $k$-ary semi-algebraic relation $E$ on $P$ is the set of $k$-tuples of points in $P$, which is determined by a finite number of polynomial equations and inequalities in $kd$ real variables. The description complexity of such a relation is at most $t$ if the number of polynomials and their degrees are all bounded by $t$. The Ramsey number $R^{d,t}_k(s,n)$ is the minimum $N$ such that any $N$-element point set $P$ in $\mathbb{R}^d$ equipped with a $k$-ary semi-algebraic relation $E$, such that $E$ has complexity at most $t$, contains $s$ members such that every $k$-tuple induced by them is in $E$, or $n$ members such that every $k$-tuple induced by them is not in $E$. We give a new upper bound for $R^{d,t}_k(s,n)$ for $k\geq 3$ and $s$ fixed. In particular, we show that for fixed integers $d,t,s$, $R^{d,t}_3(s,n) \leq 2^{n^{o(1)}},$ establishing a subexponential upper bound on $R^{d,t}_3(s,n)$. This improves the previous bound of $2^{n^C}$ due to Conlon, Fox, Pach, Sudakov, and Suk, where $C$ is a very large constant depending on $d,t,$ and $s$. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in $\mathbb{R}^d$. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results

Two segment classes with Hamiltonian visibility graphs

AbstractWe prove that the endpoint visibility graph of a set of disjoint segments that satisfy one of two restrictions, always contains a simple Hamiltonian circuit. The first restriction defines the class of independent segments: the line containing each segment misses all the other segments. The second restriction specifies unit lattice segments: unit length segments whose endpoints have integer coordinates

Higher-order Erdos--Szekeres theorems

Let P=(p_1,p_2,...,p_N) be a sequence of points in the plane, where p_i=(x_i,y_i) and x_1<x_2<...<x_N. A famous 1935 Erdos--Szekeres theorem asserts that every such P contains a monotone subsequence S of $\sqrt N$ points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of $\Omega(\log N)$ points. Monotonicity is a property determined by pairs of points, and convexity concerns triples of points. We propose a generalization making both of these theorems members of an infinite family of Ramsey-type results. First we define a (k+1)-tuple $K\subseteq P$ to be positive if it lies on the graph of a function whose kth derivative is everywhere nonnegative, and similarly for a negative (k+1)-tuple. Then we say that $S\subseteq P$ is kth-order monotone if its (k+1)-tuples are all positive or all negative. We investigate quantitative bound for the corresponding Ramsey-type result (i.e., how large kth-order monotone subsequence can be guaranteed in every N-point P). We obtain an $\Omega(\log^{(k-1)}N)$ lower bound ((k-1)-times iterated logarithm). This is based on a quantitative Ramsey-type theorem for what we call transitive colorings of the complete (k+1)-uniform hypergraph; it also provides a unified view of the two classical Erdos--Szekeres results mentioned above. For k=3, we construct a geometric example providing an $O(\log\log N)$ upper bound, tight up to a multiplicative constant. As a consequence, we obtain similar upper bounds for a Ramsey-type theorem for order-type homogeneous subsets in R^3, as well as for a Ramsey-type theorem for hyperplanes in R^4 recently used by Dujmovic and Langerman.Comment: Contains a counter example of Gunter Rote which gives a reply for the problem number 5 in the previous versions of this pape

A bipartite analogue of Dilworth's theorem for multiple partial orders

AbstractLet r be a fixed positive integer. It is shown that, given any partial orders <1, …, <r on the same n-element set P, there exist disjoint subsets A,B⊂P, each with at least n1−o(1) elements, such that one of the following two conditions is satisfied: (1) there is an i(1≤i≤r) such that every element of A is larger than every element of B in the partial order <i, or (2) no element of A is comparable with any element of B in any of the partial orders <1, …, <r. As a corollary, we obtain that any family C of n convex compact sets in the plane has two disjoint subfamilies A,B⊂C, each with at least n1−o(1) members, such that either every member of A intersects all members of B, or no member of A intersects any member of B