arXiv subjects

Felix Christian Clemen

Publications and source records attributed to Felix Christian Clemen.

32 records · Page 2Linked to original sources

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.

math.CO↗

arXiv2022

Non-degenerate Hypergraphs with Exponentially Many Extremal Constructions

For every integer $t \ge 0$, denote by $F_5^t$ the hypergraph on vertex set $\{1,2,\ldots, 5+t\}$ with hyperedges $\{123,124\} \cup \{34k : 5 \le k \le 5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t\ge 0$ and sufficiently large $n$ and characterize the extremal $F_5^t$-free hypergraphs. In particular, if $n$ satisfies certain divisibility conditions, then the extremal $F_5^t$-free hypergraphs are exactly the balanced complete tripartite hypergraphs with additional hyperedges inside each of the three parts $(V_1,V_2,V_3)$ in the partition; each part $V_i$ spans a $(|V_i|,3,2,t)$-design. This generalizes earlier work of Frankl and Füredi on the Turán number of $F_5:=F_5^0$. Our results extend a theory of Erdős and Simonovits about the extremal constructions for certain fixed graphs. In particular, the hypergraphs $F_5^{6t}$, for $t\geq 1$, are the first examples of hypergraphs with exponentially many extremal constructions and positive Turán density.

math.CO↗

arXiv2022

10 Problems for Partitions of Triangle-free Graphs

We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erdős' Sparse Half Conjecture. Among others we prove the following variant of it: For every sufficiently large even integer $n$ the following holds. Every triangle-free graph on $n$ vertices has a partition $V(G)=A\cup B$ with $|A|=|B|=n/2$ such that $e(G[A])+e(G[B])\leq n^2/16$. This result is sharp since the complete bipartite graph with class sizes $3n/4$ and $n/4$ achieves equality, when $n$ is a multiple of 4. Additionally, we discuss similar problems for $K_4$-free graphs.

math.CO↗

arXiv2022

Chasing the Threshold Bias of the 3-AP Game

In a Maker-Breaker game there are two players, Maker and Breaker, where Maker wins if they create a specified structure while Breaker wins if they prevent Maker from winning indefinitely. A $3$-term arithmetic progression, or $3$-AP, is a sequence of three distinct integers $a, b, c$ such that $b-a = c-b$. The $3$-AP game is a biased Maker-Breaker game played on $[n]$ where every round Breaker selects $q$ unclaimed integers for every Maker's one integer. Maker is trying to select points such that they have a $3$-AP and Breaker is trying to prevent this. The main question of interest is determining the threshold bias $q^*(n)$, that is the minimum value of $q=q(n)$ for which Breaker has a winning strategy. Kusch, Rué, Spiegel and Szabó initially asked this question and proved $\sqrt{n/12-1/6}\leq q^*(n)\leq \sqrt{3n}$. We find new strategies for both Maker and Breaker which improve the existing bounds to \[ (1+o(1))\sqrt{\frac{n}{5.6}} \leq q^*(n) \leq \sqrt{2n} +O(1). \]

math.CO↗

arXiv2021

Solving Turán's Tetrahedron Problem for the $\ell_2$-Norm

Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron $K_4^3$. This is equivalent to determining the maximum $\ell_1$-norm of the codegree vector of a $K_4^3$-free $n$-vertex $3$-uniform hypergraph. We introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum, $\text{co}_2(G)$, of a $3$-uniform hypergraph $G$ is the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $xy$, or in other words, the square of the $\ell_2$-norm of the codegree vector of the pairs of vertices. We define $\text{exco}_2(n,H)$ to be the maximum $\text{co}_2(G)$ over all $H$-free $n$-vertex $3$-uniform hypergraphs $G$. We use flag algebra computations to determine asymptotically the codegree squared extremal number for $K_4^3$ and $K_5^3$ and additionally prove stability results. In particular, we prove that the extremal $K_4^3$-free hypergraphs in $\ell_2$-norm have approximately the same structure as one of the conjectured extremal hypergraphs for Turán's conjecture. Further, we prove several general properties about $\text{exco}_2(n,H)$ including the existence of a scaled limit, blow-up invariance and a supersaturation result.

math.CO↗

arXiv2021

A Strengthening of the Erdős-Szekeres Theorem

The Erdős-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$ edges or a blue copy of the monotone increasing path with $s$ edges. Although $rs + 1$ is the minimum number of vertices needed for this result, not all edges of $K_{rs+1}$ are necessary. We characterize the subgraphs of $K_{rs+1}$ with this coloring property as follows: they are exactly the subgraphs that contain all the edges of a graph we call the circus tent graph $CT(r,s)$. Additionally, we use similar proof techniques to improve upon some of the bounds on the online ordered size Ramsey number of a path given by Pérez-Giménez, Pralat, and West.

math.CO↗

arXiv2021

Counting $r$-graphs without forbidden configurations

One of the major problems in combinatorics is to determine the number of $r$-uniform hypergraphs ($r$-graphs) on $n$ vertices which are free of certain forbidden structures. This problem dates back to the work of Erdős, Kleitman and Rothschild, who showed that the number of $K_r$-free graphs on $n$ vertices is $2^{\text{ex}(n,K_r)+o(n^2)}$. Their work was later extended to forbidding graphs as induced subgraphs by Prömel and Steger. Here, we consider one of the most basic counting problems for $3$-graphs. Let $E_1$ be the $3$-graph with $4$ vertices and $1$ edge. What is the number of induced $\{K_4^3,E_1\}$-free $3$-graphs on $n$ vertices? We show that the number of such $3$-graphs is of order $n^{Θ(n^2)}$. More generally, we determine asymptotically the number of induced $\mathcal{F}$-free $3$-graphs on $n$ vertices for all families $\mathcal{F}$ of $3$-graphs on $4$ vertices. We also provide upper bounds on the number of $r$-graphs on $n$ vertices which do not induce $i \in L$ edges on any set of $k$ vertices, where $L \subseteq \big \{0,1,\ldots,\binom{k}{r} \big\}$ is a list which does not contain $3$ consecutive integers in its complement. Our bounds are best possible up to a constant multiplicative factor in the exponent when $k = r+1$. The main tool behind our proof is counting the solutions of a constraint satisfaction problem.

math.CO↗

arXiv2021

On stability of the Erdős-Rademacher Problem

Mantel's theorem states that every $n$-vertex graph with $\lfloor \frac{n^2}{4} \rfloor +t$ edges, where $t>0$, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the Erdős-Rademacher problem. Lovász and Simonovits proved that there are at least $t\lfloor n/2 \rfloor$ triangles in each of those graphs. Katona and Xiao considered the same problem under the additional condition that there are no $s-1$ vertices covering all triangles. They settled the case $t=1$ and $s=2$. Solving their conjecture, we determine the minimum number of triangles for every fixed pair of $s$ and $t$, when $n$ is sufficiently large. Additionally, solving another conjecture of Katona and Xiao, we extend the theory for considering cliques instead of triangles.

math.CO↗

arXiv2021

Max Cuts in Triangle-free Graphs

A well-known conjecture by Erdős states that every triangle-free graph on $n$ vertices can be made bipartite by removing at most $n^2/25$ edges. This conjecture was known for graphs with edge density at least $0.4$ and edge density at most $0.172$. Here, we will extend the edge density for which this conjecture is true; we prove the conjecture for graphs with edge density at most $0.2486$ and for graphs with edge density at least $0.3197$. Further, we prove that every triangle-free graph can be made bipartite by removing at most $n^2/23.5$ edges improving the previously best bound of $n^2/18$.

math.CO↗

arXiv2021

Maximum Number of Almost Similar Triangles in the Plane

A triangle $T'$ is $\varepsilon$-similar to another triangle $T$ if their angles pairwise differ by at most $\varepsilon$. Given a triangle $T$, $\varepsilon>0$ and $n\in\mathbb{N}$, Bárány and Füredi asked to determine the maximum number of triangles $h(n,T,\varepsilon)$ being $\varepsilon$-similar to $T$ in a planar point set of size $n$. We show that for almost all triangles $T$ there exists $\varepsilon=\varepsilon(T)>0$ such that $h(n,T,\varepsilon)=n^3/24 (1+o(1))$. Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof.

math.CO↗

arXiv2020

Flexibility of Planar Graphs -- Sharpening the Tools to Get Lists of Size Four

A graph where each vertex $v$ has a list $L(v)$ of available colors is $L$-colorable if there is a proper coloring such that the color of $v$ is in $L(v)$ for each $v$. A graph is $k$-choosable if every assignment $L$ of at least $k$ colors to each vertex guarantees an $L$-coloring. Given a list assignment $L$, an $L$-request for a vertex $v$ is a color $c\in L(v)$. In this paper, we look at a variant of the widely studied class of precoloring extension problems from [Z. Dvořák, S. Norin, and L. Postle: List coloring with requests. J. Graph Theory 2019], wherein one must satisfy "enough", as opposed to all, of the requested set of precolors. A graph $G$ is $\varepsilon$-flexible for list size $k$ if for any $k$-list assignment $L$, and any set $S$ of $L$-requests, there is an $L$-coloring of $G$ satisfying an $\varepsilon$-fraction of the requests in $S$. It is conjectured that planar graphs are $\varepsilon$-flexible for list size $5$, yet it is proved only for list size $6$ and for certain subclasses of planar graphs. We give a stronger version of the main tool used in the proofs of the aforementioned results. By doing so, we improve upon a result by Masařík and show that planar graphs without $K_4^-$ are $\varepsilon$-flexible for list size $5$. We also prove that planar graphs without $4$-cycles and $3$-cycle distance at least 2 are $\varepsilon$-flexible for list size $4$. Finally, we introduce a new (slightly weaker) form of $\varepsilon$-flexibility where each vertex has exactly one request. In that setting, we provide a stronger tool and we demonstrate its usefulness to further extend the class of graphs that are $\varepsilon$-flexible for list size $5$.

cs.DM↗

arXiv2019

Making $K_{r+1}$-Free Graphs $r$-partite

The Erdős-Simonovits stability theorem states that for all ε>0 there exists α>0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - αn^2, then one can remove εn^2 edges from G to obtain an r-partite graph. Füredi gave a short proof that one can choose α=ε. We give a bound for the relationship of αand \varepsilon which is asymptotically sharp as ε\to 0.

math.CO↗

arXiv2019

Ordered Size Ramsey Number of Paths

An ordered graph is a simple graph with an ordering on its vertices. Define the ordered path $P_n$ to be the monotone increasing path with $n$ edges. The ordered size Ramsey number $\tilde{r}(P_r,P_s)$ is the minimum number $m$ for which there exists an ordered graph $H$ with $m$ edges such that every two-coloring of the edges of $H$ contains a red copy of $P_r$ or a blue copy of $P_s$. For $2\leq r\leq s$, we show $\frac{1}{8}r^2s\leq \tilde{r}(P_r,P_s)\leq Cr^2s(\log s)^3$, where $C>0$ is an absolute constant. This problem is motivated by the recent results of Bucić-Letzter-Sudakov and Letzter-Sudakov for oriented graphs.

math.CO↗

arXiv2019

The Ramsey Number of Fano Plane Versus Tight Path

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer $N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$ Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to determine all $H$ that are $\mathbb{F}$-good. In this short paper we make progress on this problem and prove that the tight path of length $n$ is $\mathbb{F}$-good.

math.CO↗