arXiv subjects

Andrew Suk

Publications and source records attributed to Andrew Suk.

At least 19 recordsLinked to original sources

arXiv2026

A parity Erd\H{o}s-Hajnal theorem for $t$-intersecting curves

For every fixed $t\ge 1$, we prove a parity analogue of the mighty Erd\H{o}s-Hajnal property for $t$-intersecting curves in the plane. Let $\mathcal B$ be a set of blue curves and $\mathcal G$ a set of green curves in the plane such that $\mathcal B\cup\mathcal G$ is a collection of $t$-intersecting curves in general position. We show that there exist subfamilies $\mathcal B'\subseteq\mathcal B$ and $\mathcal G'\subseteq\mathcal G$ such that $|\mathcal B'|\geq \varepsilon|\mathcal B|$ and $|\mathcal G'|\geq \varepsilon|\mathcal G|$, where $\varepsilon>0$ depends only on $t$, such that either every pair in $\mathcal B'\times\mathcal G'$ intersects an even number of times or every such pair intersects an odd number of times. For $t=1$, this recovers the theorem of Fox, Pach, and Suk for pseudo-segments. As an application, we show that every $n$-vertex topological graph with edges forming a $t$-intersecting family and with no $k$ edges that pairwise cross an odd number of times has at most $n(\log n)^{O_t(\log k)}$ edges.

math.CO↗

arXiv2026

On the maximum number of $k$-holes in point sets with no $(k + 1)$-hole

The classical problem of Erd\H{o}s asks for the minimum number of empty convex $k$-gons determined by an $n$-element point set in the plane. The celebrated empty hexagon theorem, proved independently by Gerken and Nicol\'as, shows that every sufficiently large planar point set contains a $6$-hole, while Horton's famous construction shows the existence of arbitrarily large point sets with no $7$-hole. In this paper, we initiate the study of the maximum number of $k$-holes in planar point sets with no $(k+1)$-hole. More precisely, for each fixed $k\geq 6$, let $h_k(n)$ be the maximum number of $k$-holes determined by a planar point set in general position, of size at most $n$, and with no $(k+1)$-hole. We prove that there are absolute constants $c_1,c_2>0$ such that $(c_1/k)^{\lfloor k/3\rfloor} n^{\lfloor k/3\rfloor} \leq h_k(n)\leq (c_2/k)^{\lceil k/2\rceil} n^{\lceil k/2\rceil}$.

math.CO↗

arXiv2026

Extremal structure in dense arrangements of $k$-intersecting curves

Let $P$ be a set of $n$ points in the plane, and let $\mathcal C$ be a collection of $n$ simple $k$-intersecting curves, meaning that every two distinct curves of $\mathcal C$ meet in at most $k$ points. A classical theorem of Pach and Sharir from 1998 gives the upper bound $I(P,\mathcal C)=O_k(n^{(3k+1)/(2k+1)})$. We prove that this bound can be improved when one excludes a complete local incidence pattern. More precisely, for any fixed integers $s>k+1\ge 2$, if there do not exist $s$ points of $P$ such that every $(k+1)$-tuple among them is contained in a distinct curve of $\mathcal C$, then $I(P,\mathcal C)=o(n^{(3k+1)/(2k+1)})$. In the special case of pseudo-segments, this extends Solymosi's theorem on dense point-line arrangements to dense arrangements of pseudo-segments.

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arXiv2025

Unavoidable patterns and plane paths in dense topological graphs

Let $C_{s,t}$ be the complete bipartite geometric graph, with $s$ and $t$ vertices on two distinct parallel lines respectively, and all $s t$ straight-line edges drawn between them. In this paper, we show that every complete bipartite simple topological graph, with parts of size $2(k-1)^4 + 1$ and $2^{k^{5k}}$, contains a topological subgraph weakly isomorphic to $C_{k,k}$. As a corollary, every $n$-vertex simple topological graph not containing a plane path of length $k$ has at most $O_k(n^{2 - 8/k^4})$ edges. When $k = 3$, we obtain a stronger bound by showing that every $n$-vertex simple topological graph not containing a plane path of length 3 has at most $O(n^{4/3})$ edges. We also prove that $x$-monotone simple topological graphs not containing a plane path of length 3 have at most a linear number of edges.

math.CO↗

arXiv2025

Immersions and Albertson's conjecture

A graph is said to contain $K_k$ (a clique of size $k$) as a weak immersion if it has $k$ vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger's conjecture: Every graph of chromatic number $k$ contains $K_k$ as a weak immersion. We prove this conjecture for graphs with at most $(1.64-o(1))k$ vertices. As an application, we make some progress on Albertson's conjecture, according to which every graph $G$ with chromatic number $k$ satisfies $cr(G) \geq cr(K_k)$. In particular, we show that the conjecture is true for all graphs of chromatic number $k$, provided that they have at most $(1.64-o(1))k$ vertices.

math.CO↗

arXiv2024

A note on the no-$(d+2)$-on-a-sphere problem

For fixed $d\geq 3$, we construct subsets of the $d$-dimensional lattice cube $[n]^d$ of size $n^{\frac{3}{d + 1} - o(1)}$ with no $d+2$ points on a sphere or a hyperplane. This improves the previously best known bound of $\Omega(n^{\frac{1}{d-1}})$ due to Thiele from 1995.

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arXiv2024

3-uniform monotone paths and multicolor Ramsey numbers

The monotone path $P_{n+2}$ is an ordered 3-uniform hypergraph whose vertex set has size $n+2$ and edge set consists of all consecutive triples. In this note, we consider the collection $\mathcal{J}_n$ of ordered 3-uniform hypergraphs named monotone paths with $n$ jumps, and we prove the following relation \begin{equation*} r(3;n) \leq R(P_{n+2},\mathcal{J}_n) \leq 4^n \cdot r(3;n), \end{equation*} where $r(3;n)$ is the multicolor Ramsey number for triangles and $R(P_{n+2},\mathcal{J}_n)$ is the hypergraph Ramsey number for $P_{n+2}$ versus any member of $\mathcal{J}_n$. In particular, whether $r(3;n)$ is exponential, which is a very old problem of Erd\H{o}s, is equivalent to whether $R(P_{n+2},\mathcal{J}_n)$ is exponential.

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arXiv2024

When are off-diagonal hypergraph Ramsey numbers polynomial?

A natural open problem in Ramsey theory is to determine those $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H, K_n^{(3)})$ grows polynomially with $n$. We make substantial progress on this question by showing that if $H$ is tightly connected or has at most two tight components, then $r(H, K_n^{(3)})$ grows polynomially if and only if $H$ is contained in an iterated blowup of an edge.

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arXiv2024

Enumeration of intersection graphs of $x$-monotone curves

A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each consisting of $n$ labelled $x$-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most $2^{O(n^{4/3}\log^2n)}$. Our proof uses a new upper bound on the number of set systems of size $m$ on a ground set of size $n$, with VC-dimension at most $d$. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.

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arXiv2024

Big line or big convex polygon

Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$, $$ (3\ell - 1) \cdot 2^{n-5} < ES_{\ell}(n) < \ell^2 \cdot 2^{n+ C\sqrt{n\log n}}.$$ A similar extension of the well-known Erd\H os--Szekeres cups-caps theorem is also proved.

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arXiv2024

A question of Erd\H{o}s and Graham on Egyptian fractions

Answering a question of Erd\H{o}s and Graham, we show that for each fixed positive rational number $x$ the number of ways to write $x$ as a sum of reciprocals of distinct positive integers each at most $n$ is $2^{(c_x + o(1))n}$ for an explicit constant $c_x$ increasing with $x$.

math.CO↗

arXiv2024

On off-diagonal hypergraph Ramsey numbers

A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{\Omega_H(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear.

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arXiv2023

A structure theorem for pseudo-segments and its applications

We prove a far-reaching strengthening of Szemer\'edi's regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such a graph can be partitioned into a bounded number of parts of roughly the same size such that almost all bipartite graphs between different pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every $n$-vertex simple topological graph with no $k$ pairwise disjoint edges has at most $n(\log n)^{O(\log k)}$ edges.

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arXiv2023

On cliques in three-dimensional dense point-line arrangements

As a variant of the celebrated Szemer\'edi--Trotter theorem, Guth and Katz proved that $m$ points and $n$ lines in $\mathbb{R}^3$ with at most $\sqrt{n}$ lines in a common plane must determine at most $O(m^{1/2}n^{3/4})$ incidences for $n^{1/2}\leq m\leq n^{3/2}$. This upper bound is asymptotically tight and has an important application in Erd\H{o}s distinct distance problem. We characterize the extremal constructions towards the Guth--Katz bound by proving that such a large dense point-line arrangement must contain a $k$-clique in general position provided $m \ll n$. This is an analog of a result by Solymosi for extremal Szemer\'edi--Trotter constructions in the plane.

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arXiv2023

On short edges in complete topological graphs

Let $h(n)$ be the minimum integer such that every complete $n$-vertex simple topological graph contains an edge that crosses at most $h(n)$ other edges. In 2009, Kyn\v{c}l and Valtr showed that $h(n) = O(n^2/\log^{1/4} n)$, and in the other direction, gave constructions showing that $h(n) = \Omega(n^{3/2})$. In this paper, we prove that $h(n) = O(n^{7/4})$. Along the way, we establish a new variant of Chazelle and Welzl's matching theorem for set systems with bounded VC-dimension, which we believe to be of independent interest.

math.CO↗

arXiv2023

Ramsey numbers of cliques versus monotone paths

One formulation of the Erdos-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on $\{1, 2, \ldots, N\}$, there exists a monochromatic red $s$-clique or a monochromatic blue increasing path $P_n$ with $n$ vertices, provided $N >(s-1)(n-1)$. %We had previously shown that a suitable generalization of this problem to quadruple systems is essentially equivalent to classical diagonal hypergraph Ramsey numbers. Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound $N>2^{c(\log n)^{s-1}}$. For the $t$th power $P_n^t$ of the ordered increasing graph path with $n$ vertices, we prove a near linear bound $c\, n(\log n)^{s-2}$ which improves the previous bound that applied to a more general class of graphs than $P_n^t$ due to Conlon-Fox-Lee-Sudakov.

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arXiv2022

Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems

Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete $n$-vertex simple topological graph generates at least $Ω(n^{1/3})$ pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on $n$ vertices drawn in the unit square generates a 4-face with area at most $O(n^{-1/3})$. Finally, we investigate a $\mathbb Z_2$ variant of Heilbronn triangle problem.

math.CO↗

arXiv2022

On higher dimensional point sets in general position

A finite point set in $\mathbb{R}^d$ is in general position if no $d + 1$ points lie on a common hyperplane. Let $\alpha_d(N)$ be the largest integer such that any set of $N$ points in $\mathbb{R}^d$, with no $d + 2$ members on a common hyperplane, contains a subset of size $\alpha_d(N)$ in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that $\alpha_2(N) < N^{5/6 + o(1)}$. In this paper, we also use the container method to obtain new upper bounds for $\alpha_d(N)$ when $d \geq 3$. More precisely, we show that if $d$ is odd, then $\alpha_d(N) < N^{\frac{1}{2} + \frac{1}{2d} + o(1)}$, and if $d$ is even, we have $\alpha_d(N) < N^{\frac{1}{2} + \frac{1}{d-1} + o(1)}$. We also study the classical problem of determining $a(d,k,n)$, the maximum number of points selected from the grid $[n]^d$ such that no $k + 2$ members lie on a $k$-flat, and improve the previously best known bound for $a(d,k,n)$, due to Lefmann in 2008, by a polynomial factor when $k$ = 2 or 3 (mod 4).

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