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Felix Christian Clemen

Publications and source records attributed to Felix Christian Clemen.

At least 19 recordsLinked to original sources
arXiv2026

The Spectrum of Triangle-free Graphs

Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \leq \frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) \leq \frac{15n}{94}< \frac{4n}{25}$. Brandt's conjecture is a subproblem of two famous conjectures of Erdős: (1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset of vertices of size $\left\lfloor\frac{n}{2}\right\rfloor$ spanning at most $n^2/50$ edges. (2) Every $n$-vertex triangle-free graph can be made bipartite by removing at most $n^2/25$ edges. In our proof we use linear algebraic methods to upper bound $q_n(G)$ by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.

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arXiv2026

Subset selection problems in planar point sets

Given a finite point set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can be extracted? In this paper, we make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$. This problem was proposed by Erdős under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$ and $O(n/s)$ when $n^{1/3}\leq s\leq n$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=Ω(\sqrt{n})$, our upper and lower bounds differ at most by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erdős, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $Ω((n/\log{s})^{1/3})$ can always be found in $P$. When $s=Θ(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

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arXiv2026

Triangle-Saturated Graphs in the Semi-Random Graph Process

The semi-random graph process is an adaptive random graph process in which an online algorithm is initially given an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. We also consider the offline version of the process in which the algorithm is given the entire sequence of random vertex choices before the selection takes place. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible. In this paper, we focus on the property of being triangle-saturated and establish upper and lower bounds on the number of rounds required to construct a triangle-saturated graph in both the online and offline versions of the process.

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arXiv2026

The number of regular simplices in higher dimensions

We study the extremal function $S^k_d(n)$, defined as the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For any fixed $d\geq2k\geq6$, we determine the asymptotic behavior of $S^k_d(n)$ up to a lower-order term. In particular, when $k=3$, we determine the exact value of $S^3_d(n)$, for all even dimensions $d\geq6$ and sufficiently large $n$. This resolves a conjecture of Erdős in a stronger form. The proof leverages techniques from hypergraph Turán theory and linear algebra.

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arXiv2026

Geometric Sidon Problems

This paper considers geometric problems of the following type: given a point set $P \subset \mathbb R^2$, one seeks a large subset avoiding a prescribed geometric configuration. Our main result states that, for any $P \subset \mathbb R^2$, there exists a subset $P' \subset P$ with $|P'| \gg |P|^{1/3}$ such that all of the distances determined by $P'$ are distinct. This improves a result of Charalambides. We make heavy use of a result of Li and Postle concerning the independence number of hypergraphs which satisfy some edge distribution conditions, as well as tools from incidence geometry.

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arXiv2026

On Tournament Anti-Sidorenko Orientations of Trees

An oriented graph $\vec{H}$ is said to be tournament anti-Sidorenko if the homomorphism density of $\vec{H}$ in any tournament $\vec{T}$ is bounded above by the homomorphism density of $\vec{H}$ in a large uniformly random tournament. We prove the following: (1) Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko. (2) An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a multiple of four. (3) Every spider with exactly three legs admits a tournament anti-Sidorenko orientation. The first result proves a conjecture posed by He, Mani, Nie, Tung and Wei. The third resolves a problem from the same paper, in fact establishing a substantially more general statement, and provides evidence in support of a conjecture of Fox, Himwich, Mani and Zhou. The second yields the first family of tournament anti-Sidorenko oriented paths which is exponentially large with respect to the number of arcs.

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arXiv2026

On multiplicities of interpoint distances

Given a set $X\subseteq\mathbb{R}^2$ of $n$ points and a distance $d>0$, the multiplicity of $d$ is the number of times the distance $d$ appears between points in $X$. Let $a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)$ denote the multiplicities of the $m$ distances determined by $X$ and let $a(X)=\left(a_1(X),\dots,a_m(X)\right)$. In this paper, we study several questions from Erdős's time regarding distance multiplicities. Among other results, we show that: (1) If $X$ is convex or ``not too convex'', then there exists a distance other than the diameter that has multiplicity at most $n$. (2) There exists a set $X \subseteq \mathbb{R}^2$ of $n$ points, such that many distances occur with high multiplicity. In particular, at least $n^{Ω(1/\log\log{n})}$ distances have superlinear multiplicity in $n$. (3) For any (not necessarily fixed) integer $1\leq k\leq\log{n}$, there exists $X\subseteq\mathbb{R}^2$ of $n$ points, such that the difference between the $k^{\text{th}}$ and $(k+1)^{\text{th}}$ largest multiplicities is at least $Ω(\frac{n\log{n}}{k})$. Moreover, the distances in $X$ with the largest $k$ multiplicities can be prescribed. (4) For every $n\in\mathbb{N}$, there exists $X\subseteq\mathbb{R}^2$ of $n$ points, not all collinear or cocircular, such that $a(X)= (n-1,n-2,\ldots,1)$. There also exists $Y\subseteq\mathbb{R}^2$ of $n$ points with pairwise distinct distance multiplicities and $a(Y) \neq (n-1,n-2,\ldots,1)$.

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arXiv2025

Maximizing Alternating Paths via Entropy

We prove that if $G$ is an $n$-vertex graph whose edges are coloured with red and blue, then the number of colour-alternating walks of length $2k+1$ with $k+1$ red edges and $k$ blue edges is at most $k^k(k+1)^{k+1}(2k+1)^{-2k-1}n^{2k+2}$. This solves a problem that was recently posed by Basit, Granet, Horsley, Kündgen and Staden. Our proof involves an application of the entropy method.

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arXiv2025

Hypergraph Turán Problems in $\ell_2$-Norm

There are various different notions measuring extremality of hypergraphs. In this survey we compare the recently introduced notion of the codegree squared extremal function with the Turán function, the minimum codegree threshold and the uniform Turán density. The codegree squared sum $\textrm{co}_2(G)$ of a $3$-uniform hypergraph $G$ is defined to be the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $x,y$. In other words, this is the square of the $\ell_2$-norm of the codegree vector. We are interested in how large $\textrm{co}_2(G)$ can be if we require $G$ to be $H$-free for some $3$-uniform hypergraph $H$. This maximum value of $\textrm{co}_2(G)$ over all $H$-free $n$-vertex $3$-uniform hypergraphs $G$ is called the codegree squared extremal function, which we denote by $\textrm{exco}_2(n,H)$. We systemically study the extremal codegree squared sum of various $3$-uniform hypergraphs using various proof techniques. Some of our proofs rely on the flag algebra method while others use more classical tools such as the stability method. In particular, we (asymptotically) determine the codegree squared extremal numbers of matchings, stars, paths, cycles, and $F_5$, the $5$-vertex hypergraph with edge set $\{123,124,345\}$. Additionally, our paper has a survey format, as we state several conjectures and give an overview of Turán densities, minimum codegree thresholds and codegree squared extremal numbers of popular hypergraphs. We intend to update the arXiv version of this paper regularly.

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arXiv2024

Crossing Numbers of Billiard Curves in the Multidimensional Box via Translation Surfaces

The billiard table is modeled as an $n$-dimensional box $[0,a_1]\times [0,a_2]\times \ldots \times [0,a_n] \subset \mathbb{R}^n$, with each side having real-valued lengths $a_i$ that are pairwise commensurable. A ball is launched from the origin in direction $d=(1,1,\ldots,1)$. The ball is reflected if it hits the boundary of the billiard table. It comes to a halt when reaching a corner. We show that the number of intersections of the billiard curve at any given point on the table is either $0$ or a power of $2$. To prove this, we use algebraic and number theoretic tools to establish a bijection between the number of intersections of the billiard curve and the number of satisfying assignments of a specific constraint satisfaction problem.

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arXiv2024

On a Traveling Salesman Problem for Points in the Unit Cube

Let $X$ be an $n$-element point set in the $k$-dimensional unit cube $[0,1]^k$ where $k \geq 2$. According to an old result of Bollobás and Meir (1992), there exists a cycle (tour) $x_1, x_2, \ldots, x_n$ through the $n$ points, such that $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} \leq c_k$, where $|x-y|$ is the Euclidean distance between $x$ and $y$, and $c_k$ is an absolute constant that depends only on $k$, where $x_{n+1} \equiv x_1$. From the other direction, for every $k \geq 2$ and $n \geq 2$, there exist $n$ points in $[0,1]^k$, such that their shortest tour satisfies $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} = 2^{1/k} \cdot \sqrt{k}$. For the plane, the best constant is $c_2=2$ and this is the only exact value known. Bollob{á}s and Meir showed that one can take $c_k = 9 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ for every $k \geq 3$ and conjectured that the best constant is $c_k = 2^{1/k} \cdot \sqrt{k}$, for every $k \geq 2$. Here we significantly improve the upper bound and show that one can take $c_k = 3 \sqrt5 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ or $c_k = 2.91 \sqrt{k} \ (1+o_k(1))$. Our bounds are constructive. We also show that $c_3 \geq 2^{7/6}$, which disproves the conjecture for $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás--Meir conjecture is proposed.

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arXiv2024

Applications of Sparse Hypergraph Colorings

Many problems in extremal combinatorics can be reduced to determining the independence number of a specific auxiliary hypergraph. We present two such problems, one from discrete geometry and one from hypergraph Turán theory. Using results on hypergraph colorings by Cooper-Mubayi and Li-Postle, we demonstrate that for those two problems the trivial lower bound on the independence number can be improved upon: Erdős, Graham, Ruzsa and Taylor asked to determine the largest size, denoted by $g(n)$, of a subset $P$ of the grid $[n]^2$ such that every pair of points in $P$ span a different slope. Improving on a lower bound by Zhang from 1993, we show that $$g(n)=Ω\left( \frac{n^{2/3} (\log \log n)^{1/3} }{ \log^{1/3}n} \right).$$ Let $H^r_3$ denote an $r$-graph with $r+1$ vertices and $3$ edges. Recently, Sidorenko proved the following lower bounds for the Turán density of this $r$-graph: $π(H^r_3)\geq r^{-2}$ for every $r$, and $π(H^r_3)\geq (1.7215 - o(1)) r^{-2}$. We present an improved asymptotic bound: $π(H^r_3)=Ω\left(r^{-2} \log^{1/2} r \right).$

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arXiv2024

Online Ramsey numbers of ordered paths and cycles

An ordered graph is a graph with a linear ordering on its vertices. The online Ramsey game for ordered graphs $G$ and $H$ is played on an infinite sequence of vertices; on each turn, Builder draws an edge between two vertices, and Painter colors it red or blue. Builder tries to create a red $G$ or a blue $H$ as quickly as possible, while Painter wants the opposite. The online ordered Ramsey number $r_o(G,H)$ is the number of turns the game lasts with optimal play. In this paper, we consider the behavior of $r_o(G,P_n)$ for fixed $G$, where $P_n$ is the monotone ordered path. We prove an $O(n \log_2n)$ bound on $r_o(G,P_n)$ for all $G$ and an $O(n)$ bound when $G$ is $3$-ichromatic; we partially classify graphs $G$ with $r_o(G,P_n) = n + O(1)$. Many of these results extend to $r_o(G,C_n)$, where $C_n$ is an ordered cycle obtained from $P_n$ by adding one edge.

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arXiv2023

Almost Congruent Triangles

Almost $50$ years ago Erdős and Purdy asked the following question: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three small clusters built around the three vertices of a fixed equilateral triangle, one gets at least $\left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n+1}{3} \right\rfloor \cdot \left\lfloor \frac{n+2}{3} \right\rfloor$ such approximate copies. In this paper we provide a matching upper bound and thereby answer their question. More generally, for every triangle $T$ we determine the maximum number of approximate congruent triangles to $T$ in a point set of size $n$. Parts of our proof are based on hypergraph Turán theory: for each point set in the plane and a triangle $T$, we construct a $3$-uniform hypergraph $\mathcal{H}=\mathcal{H}(T)$, which contains no hypergraph as a subgraph from a family of forbidden hypergraphs $\mathcal{F}=\mathcal{F}(T)$. Our upper bound on the number of edges of $\mathcal{H}$ will determine the maximum number of triangles that are approximate congruent to $T$.

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arXiv2023

A note on balanced edge-colorings avoiding rainbow cliques of size four

A balanced edge-coloring of the complete graph is an edge-coloring such that every vertex is incident to each color the same number of times. In this short note, we present a construction of a balanced edge-coloring with six colors of the complete graph on $n=13^k$ vertices, for every positive integer $k$, with no rainbow $K_4$. This solves a problem by Erdős and Tuza.

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arXiv2022

Rainbow Subgraphs in Edge-colored Complete Graphs -- Answering two Questions by Erdős and Tuza

An edge-coloring of a complete graph with a set of colors $C$ is called completely balanced if any vertex is incident to the same number of edges of each color from $C$. Erdős and Tuza asked in $1993$ whether for any graph $F$ on $\ell$ edges and any completely balanced coloring of any sufficiently large complete graph using $\ell$ colors contains a rainbow copy of $F$. This question was restated by Erdős in his list of ``Some of my favourite problems on cycles and colourings''. We answer this question in the negative for most cliques $F=K_q$ by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph $F$.

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arXiv2022

Unavoidable order-size pairs in hypergraphs -- positive forcing density

Erdős, Füredi, Rothschild and Sós initiated a study of classes of graphs that forbid every induced subgraph on a given number $m$ of vertices and number $f$ of edges. Extending their notation to $r$-graphs, we write $(n,e) \to_r (m,f)$ if every $r$-graph $G$ on $n$ vertices with $e$ edges has an induced subgraph on $m$ vertices and $f$ edges. The \emph{forcing density} of a pair $(m,f)$ is $$ σ_r(m,f) =\left. \limsup\limits_{n \to \infty} \frac{|\{e : (n,e) \to_r (m,f)\}|}{\binom{n}{r}} \right. .$$ In the graph setting it is known that there are infinitely many pairs $(m, f)$ with positive forcing density. Weber asked if there is a pair of positive forcing density for $r\geq 3$ apart from the trivial ones $(m, 0)$ and $(m, \binom{m}{r})$. Answering her question, we show that $(6,10)$ is such a pair for $r=3$ and conjecture that it is the unique such pair. Further, we find necessary conditions for a pair to have positive forcing density, supporting this conjecture.

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