On the Erd\H{o}s-Tuza-Valtr Conjecture

The Erd\H{o}s-Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that any set of more than $\sum_{i = n - b}^{a - 2} \binom{n - 2}{i}$ points in a plane either contains the vertices of a convex $n$-gon, $a$ points lying on a concave downward curve, or $b$ points lying on a concave upward curve. They also showed that the generalization is actually equivalent to the Erd\H{o}s-Szekeres conjecture. We prove the first new case of the Erd\H{o}s-Tuza-Valtr conjecture since the original 1935 paper of Erd\H{o}s and Szekeres. Namely, we show that any set of $\binom{n-1}{2} + 2$ points in the plane with no three points on a line and no two points sharing the same $x$-coordinate either contains 4 points lying on a concave downward curve or the vertices of a convex $n$-gon.Comment: 16 pages, 8 figure

Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers c, s ≥ 1, let M3 (c, s) be the least integer such that any set of at least M3 (c, s) points in the plane, no three on a line and colored with c colors, contains a monochromatic triangle with at most s interior points. The case s = 0 , which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that M3 (1, 0) = 3, M3 (2, 0) = 9, and M3 (c, 0) = ∞, for c ≥ 3. In this paper we extend these results when c ≥ 2 and s ≥ 1. We prove that the least integer λ3 (c) such that M3 (c, λ3 (c)) \u3c ∞ satisfies: ⌊(c-1)/2⌋ ≤ λ3 (c) ≤ c - 2, where c ≥ 2. Moreover, the exact values of M3 (c, s) are determined for small values of c and s. We also conjecture that λ3 (4) = 1, and verify it for sufficiently large Horton sets

Convex Polygons in Cartesian Products

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors

Convex polygons in Cartesian products

We study several problems concerning convex polygons whose vertices lie in aCartesian product of two sets of n real numbers (for short, grid). First, we prove that everysuch grid contains Ω(log n) points in convex position and that this bound is tight up to aconstant factor. We generalize this result to d dimensions (for a fixed d ∈ N), and obtaina tight lower bound of Ω(logd−1 n) for the maximum number of points in convex positionin a d-dimensional grid. Second, we present polynomial-time algorithms for computing thelongest x- or y-monotone convex polygonal chain in a grid that contains no two points withthe same x- or y-coordinate. We show that the maximum size of a convex polygon with suchunique coordinates can be efficiently approximated up to a factor of 2. Finally, we presentexponential bounds on the maximum number of point sets in convex position in such grids,and for some restricted variants. These bounds are tight up to polynomial factors

Induced Ramsey-type results and binary predicates for point sets

Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Ne\v{s}et\v{r}il and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey. As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $\Gamma$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.Comment: 22 pages, 3 figures, final version, minor correction

Semi-algebraic and semi-linear Ramsey numbers

An $r$-uniform hypergraph $H$ is semi-algebraic of complexity $\mathbf{t}=(d,D,m)$ if the vertices of $H$ correspond to points in $\mathbb{R}^{d}$, and the edges of $H$ are determined by the sign-pattern of $m$ degree-$D$ polynomials. Semi-algebraic hypergraphs of bounded complexity provide a general framework for studying geometrically defined hypergraphs. The much-studied semi-algebraic Ramsey number $R_{r}^{\mathbf{t}}(s,n)$ denotes the smallest $N$ such that every $r$-uniform semi-algebraic hypergraph of complexity $\mathbf{t}$ on $N$ vertices contains either a clique of size $s$, or an independent set of size $n$. Conlon, Fox, Pach, Sudakov, and Suk proved that R_{r}^{\mathbf{t}}(n,n)<\mbox{tw}_{r-1}(n^{O(1)}), where \mbox{tw}_{k}(x) is a tower of 2's of height $k$ with an $x$ on the top. This bound is also the best possible if $\min\{d,D,m\}$ is sufficiently large with respect to $r$. They conjectured that in the asymmetric case, we have $R_{3}^{\mathbf{t}}(s,n)<n^{O(1)}$ for fixed $s$. We refute this conjecture by showing that $R_{3}^{\mathbf{t}}(4,n)>n^{(\log n)^{1/3-o(1)}}$ for some complexity $\mathbf{t}$. In addition, motivated by results of Bukh-Matou\v{s}ek and Basit-Chernikov-Starchenko-Tao-Tran, we study the complexity of the Ramsey problem when the defining polynomials are linear, that is, when $D=1$. In particular, we prove that $R_{r}^{d,1,m}(n,n)\leq 2^{O(n^{4r^2m^2})}$, while from below, we establish $R^{1,1,1}_{r}(n,n)\geq 2^{\Omega(n^{\lfloor r/2\rfloor-1})}$.Comment: 23 pages, 1 figur