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Mathias Schacht

Publications and source records attributed to Mathias Schacht.

54 records · Page 3Linked to original sources
arXiv2016

Ramsey properties of random graphs and Folkman numbers

For two graphs, $G$ and $F$, and an integer $r\ge2$ we write $G\rightarrow (F)_r$ if every $r$-coloring of the edges of $G$ results in a monochromatic copy of $F$. In 1995, the first two authors established a threshold edge probability for the Ramsey property $G(n,p)\to (F)_r$, where $G(n,p)$ is a random graph obtained by including each edge of the complete graph on $n$ vertices, independently, with probability $p$. The original proof was based on the regularity lemma of Szemerédi and this led to tower-type dependencies between the involved parameters. Here, for $r=2$, we provide a self-contained proof of a quantitative version of the Ramsey threshold theorem with only double exponential dependencies between the constants. As a corollary we obtain a double exponential upper bound on the 2-color Folkman numbers. By a different proof technique, a similar result was obtained independently by Conlon and Gowers.

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arXiv2016

A note on induced Ramsey numbers

The induced Ramsey number $r_{\mathrm{ind}}(F)$ of a $k$-uniform hypergraph $F$ is the smallest natural number $n$ for which there exists a $k$-uniform hypergraph $G$ on $n$ vertices such that every two-coloring of the edges of $G$ contains an induced monochromatic copy of $F$. We study this function, showing that $r_{\mathrm{ind}}(F)$ is bounded above by a reasonable power of $r(F)$. In particular, our result implies that $r_{\mathrm{ind}}(F) \leq 2^{2^{ct}}$ for any $3$-uniform hypergraph $F$ with $t$ vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on an application of the hypergraph container method.

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arXiv2016

Embedding tetrahedra into quasirandom hypergraphs

We investigate extremal problems for quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is $(d,η)$-quasirandom if for any subset $X\subseteq V$ and every set of pairs $P\subseteq V\times V$ the number of pairs $(x,(y,z))\in X\times P$ with $\{x,y,z\}$ being a hyperedge of $H$ is in the interval $d|X||P|\pmη|V|^3$. We show that for any $\varepsilon>0$ there exists $η>0$ such that every sufficiently large $(1/2+\varepsilon,η)$-quasirandom hypergraph contains a tetrahedron, i.e., four vertices spanning all four hyperedges. A known random construction shows that the density $1/2$ is best possible. This result is closely related to a question of Erdős, whether every weakly quasirandom $3$-uniform hypergraph $H$ with density bigger than $1/2$, i.e., every large subset of vertices induces a hypergraph with density bigger than $1/2$, contains a tetrahedron.

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arXiv2016

Extremal results for random discrete structures

We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemerédi's theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Turán-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, Łuczak, and Rödl for Turán-type problems in random graphs. Similar results were obtained by Conlon and Gowers.

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arXiv2016

Packing minor-closed families of graphs into complete graphs

Motivated by a conjecture of Gyárfás, recently Böttcher, Hladký, Piguet, and Taraz showed that every collection $T_1,\dots,T_t$ of trees on $n$ vertices with $\sum_{i=1}^te(T_i)\leq \binom{n}{2}$ and with bounded maximum degree, can be packed into the complete graph on $(1+o(1))n$ vertices. We generalise this result where we relax the restriction of packing families of trees to families of graphs of any given non-trivial minor-closed class of graphs.

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arXiv2016

On the structure of graphs with given odd girth and large minimum degree

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andrásfai, Erd\H os, and Sós implies that every $n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than $\frac{2}{2k+1}n$ must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the cases $k=2$ and $3$, we show that every $n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than $\frac{3}{4k}n$ is homomorphic to the cycle of length $2k+1$. This is best possible in the sense that there are graphs with minimum degree $\frac{3}{4k}n$ and odd girth $2k+1$ which are not homomorphic to the cycle of length $2k+1$. Similar results were obtained by Brandt and Ribe-Baumann.

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arXiv2016

Extremal results for odd cycles in sparse pseudorandom graphs

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $Γ$ the generalized Turán density $π_F(Γ)$ denotes the density of a maximum subgraph of $Γ$, which contains no copy of~$F$. Extending classical Turán type results for odd cycles, we show that $π_{F}(Γ)=1/2$ provided $F$ is an odd cycle and $Γ$ is a sufficiently pseudorandom graph. In particular, for $(n,d,λ)$-graphs $Γ$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-λ,λ]$, our result holds for odd cycles of length $\ell$, provided \[ λ^{\ell-2}\ll \frac{d^{\ell-1}}n\log(n)^{-(\ell-2)(\ell-3)}\,. \] Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabó, and Vu, who addressed the case when $F$ is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free $(n,d,λ)$-graphs) shows that our assumption on $Γ$ is best possible up to the polylog-factor for every odd $\ell\geq 5$.

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arXiv2016

An exponential-type upper bound for Folkman numbers

For given integers $k$ and $r$, the Folkman number $f(k;r)$ is the smallest number of vertices in a graph $G$ which contains no clique on $k+1$ vertices, yet for every partition of its edges into $r$ parts, some part contains a clique of order $k$. The existence (finiteness) of Folkman numbers was established by Folkman (1970) for $r=2$ and by Nešetřil and Rödl (1976) for arbitrary $r$, but these proofs led to very weak upper bounds on $f(k;r)$. Recently, Conlon and Gowers and independently the authors obtained a doubly exponential bound on $f(k;2)$. Here, we establish a further improvement by showing an upper bound on $f(k;r)$ which is exponential in a polynomial function of $k$ and $r$. This is comparable to the known lower bound $2^{Ω(rk)}$. Our proof relies on a recent result of Saxton and Thomason (2015) (or, alternatively, on a recent result of Balogh, Morris, and Samotij (2015)) from which we deduce a quantitative version of Ramsey's theorem in random graphs.

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arXiv2016

Counting results for sparse pseudorandom hypergraphs II

We present a variant of a universality result of Rödl [On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), no. 1-2, 125-134] for sparse, $3$-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom'' uniform hypergraphs, which is proved in the companion paper [Counting results for sparse pseudorandom hypergraphs I].

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arXiv2016

Counting results for sparse pseudorandom hypergraphs I

We establish a so-called counting lemma that allows embeddings of certain linear uniform hypergraphs into sparse pseudorandom hypergraphs, generalizing a result for graphs [Embedding graphs with bounded degree in sparse pseudorandom graphs, Israel J. Math. 139 (2004), 93-137]. Applications of our result are presented in the companion paper [Counting results for sparse pseudorandom hypergraphs II].

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arXiv2016

Ramsey numbers for bipartite graphs with small bandwidth

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.

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arXiv2016

Upper bounds on probability thresholds for asymmetric Ramsey properties

Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge probability $p$) satisfies with probability $1-o(1)$ that every red-blue-coloring of its edges contains a red copy of $G$ or a blue copy of $H$. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes the exact order of magnitude of the threshold for the case in which $G$ and $H$ are complete graphs of arbitrary size. In our proof we present an alternative to the so-called deletion method, which was introduced by Rödl and Ruciński in their study of symmetric Ramsey properties of random graphs (i.e. the case $G=H$), and has been used in many proofs of similar results since then.

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arXiv2016

On the structure of dense graphs with fixed clique number

We study structural properties of graphs with fixed clique number and high minimum degree. In particular, we show that there exists a function $L=L(r,\varepsilon)$, such that every $K_r$-free graph $G$ on $n$ vertices with minimum degree at least $(\frac{2r-5}{2r-3}+\varepsilon)n$ is homomorphic to a $K_r$-free graph on at most $L$ vertices. It is known that the required minimum degree condition is approximately best possible for this result. For $r=3$ this result was obtained by Łuczak [On the structure of triangle-free graphs of large minimum degree, Combinatorica 26 (2006), no. 4, 489-493] and, more recently, Goddard and Lyle [Dense graphs with small clique number, J. Graph Theory 66 (2011), no. 4, 319-331] deduced the general case from Łuczak's result. Łuczak's proof was based on an application of Szemerédi's regularity lemma and, as a consequence, it only gave rise to a tower-type bound on $L(3,\varepsilon)$. The proof presented here replaces the application of the regularity lemma by a probabilistic argument, which yields a bound for $L(r,\varepsilon)$ that is doubly exponential in poly($\varepsilon$).

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arXiv2013

Extremal results in random graphs

According to Paul Erdős [Some notes on Turán's mathematical work, J. Approx. Theory 29 (1980), page 4] it was Paul Turán who "created the area of extremal problems in graph theory". However, without a doubt, Paul Erdős popularized extremal combinatorics, by his many contributions to the field, his numerous questions and conjectures, and his influence on discrete mathematicians in Hungary and all over the world. In fact, most of the early contributions in this field can be traced back to Paul Erdős, Paul Turán, as well as their collaborators and students. Paul Erdős also established the probabilistic method in discrete mathematics, and in collaboration with Alfréd Rényi, he started the systematic study of random graphs. We shall survey recent developments at the interface of extremal combinatorics and random graph theory.

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arXiv2010

Untangling planar graphs from a specified vertex position - Hard cases

Given a planar graph $G$, we consider drawings of $G$ in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding $π$ of the vertex set of $G$ into the plane. We prove that a wheel graph $W_n$ admits a drawing $π$ such that, if one wants to eliminate edge crossings by shifting vertices to new positions in the plane, then at most $(2+o(1))\sqrt n$ of all $n$ vertices can stay fixed. Moreover, such a drawing $π$ exists even if it is presupposed that the vertices occupy any prescribed set of points in the plane. Similar questions are discussed for other families of planar graphs.

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arXiv2010

Note on bipartite graph tilings

Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our result extends the work of Zhao, who determined the minimum degree threshold which guarantees that a bipartite graph has a K_{s,s}-factor.

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