Mathias Schacht
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Mathias Schacht has a total of 150 co-authors. Here are the top co-authors by number of shared publications:
- Vojtěch Rödl
Emory Univ, Atlanta, GA 30322 USA
29 shared publications - Christian Reiher
Universität Hamburg, Germany
21 shared publications Vojtěch Rödl
Emory University
13 shared publications- Yoshiharu Kohayakawa
Instituto de Matemática e Estatística Universidade de Sao Paulo Sao Paulo Brazil
12 shared publications - Yury Person
Technische Universität Ilmenau
12 shared publications - Andrzej Ruciński
Department of Discrete Mathematics Adam Mickiewicz University Poznań Poland
10 shared publications
Latest Publications
- Open Access
- 10 July 2026
- Published by Elsevier in Journal of Combinatorial Theory, Series B
- Vol. 180, 151-163
Canonical Ramsey numbers for partite hypergraphs
- 10 July 2026
- Published by Elsevier in Journal of Combinatorial Theory, Series B
- Vol. 180, 151-163
Open AccessCanonical Ramsey numbers for partite hypergraphs
- Open Access
- 27 February 2026
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 14
Sharp thresholds for Ramsey properties
In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as noncolourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this noncolourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices. Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph H in r colours has a sharp threshold in $G_{n,p}$ , for all $r \geqslant 2$ and all H in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden’s theorem and Schur’s theorem, also in any number of colours.- 27 February 2026
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 14
Open AccessSharp thresholds for Ramsey properties
In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as noncolourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this noncolourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices. Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph H in r colours has a sharp threshold in $G_{n,p}$ , for all $r \geqslant 2$ and all H in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden’s theorem and Schur’s theorem, also in any number of colours. - 23 October 2025
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
Counting odd cycles in sparse pseudorandom graphs
We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let Γ \Gamma be an ( n , d , λ ) (n,d,\lambda ) -graph, i.e., n n -vertex, d d -regular graphs with all nontrivial eigenvalues in the interval [ − λ , λ ] [-\lambda ,\lambda ] . Krivelevich, Lee, and Sudakov [SIAM J. Discrete Math. 24 (2010), pp. 1–16] conjectured that, whenever λ 2 k − 1 ≪ d 2 k / n \lambda ^{2k-1}\ll d^{2k}/n , every subgraph G G of Γ \Gamma with ( 1 / 2 + o ( 1 ) ) e ( Γ ) (1/2+o(1))e(\Gamma ) edges contains an odd cycle C 2 k + 1 C_{2k+1} . Aigner-Horev, Hàn, and the third author [Combinatorica 34 (2014), pp. 379–406] proved a weaker statment by allowing an extra polylogarithmic factor in the assumption λ 2 k − 1 ≪ d 2 k / n \lambda ^{2k-1}\ll d^{2k}/n , but we completely remove it and hence settle the conjecture. This also generalises Sudakov, Szabó, and Vu’s Turán-type theorem for triangles. Secondly, we obtain a Ramsey multiplicity result for odd cycles. Namely, in the same range of parameters, we prove that every 2-edge-colouring of Γ \Gamma contains at least ( 1 − o ( 1 ) ) 2 − 2 k d 2 k + 1 (1-o(1))2^{-2k}d^{2k+1} monochromatic copies of C 2 k + 1 C_{2k+1} . Both results are asymptotically best possible by Alon and Kahale’s construction of C 2 k + 1 C_{2k+1} -free pseudorandom graphs.
- 23 October 2025
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
Counting odd cycles in sparse pseudorandom graphs
We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let Γ \Gamma be an ( n , d , λ ) (n,d,\lambda ) -graph, i.e., n n -vertex, d d -regular graphs with all nontrivial eigenvalues in the interval [ − λ , λ ] [-\lambda ,\lambda ] . Krivelevich, Lee, and Sudakov [SIAM J. Discrete Math. 24 (2010), pp. 1–16] conjectured that, whenever λ 2 k − 1 ≪ d 2 k / n \lambda ^{2k-1}\ll d^{2k}/n , every subgraph G G of Γ \Gamma with ( 1 / 2 + o ( 1 ) ) e ( Γ ) (1/2+o(1))e(\Gamma ) edges contains an odd cycle C 2 k + 1 C_{2k+1} . Aigner-Horev, Hàn, and the third author [Combinatorica 34 (2014), pp. 379–406] proved a weaker statment by allowing an extra polylogarithmic factor in the assumption λ 2 k − 1 ≪ d 2 k / n \lambda ^{2k-1}\ll d^{2k}/n , but we completely remove it and hence settle the conjecture. This also generalises Sudakov, Szabó, and Vu’s Turán-type theorem for triangles. Secondly, we obtain a Ramsey multiplicity result for odd cycles. Namely, in the same range of parameters, we prove that every 2-edge-colouring of Γ \Gamma contains at least ( 1 − o ( 1 ) ) 2 − 2 k d 2 k + 1 (1-o(1))2^{-2k}d^{2k+1} monochromatic copies of C 2 k + 1 C_{2k+1} . Both results are asymptotically best possible by Alon and Kahale’s construction of C 2 k + 1 C_{2k+1} -free pseudorandom graphs.
- Open Access
Unavoidable subgraphs in Ramsey graphs
We study subgraphs that appear in large Ramsey graphs for a given graph $F$. The recent girth Ramsey theorem of the first two authors asserts that there are Ramsey graphs such that all small subgraphs are `forests of copies of $F$' amalgamated on vertices and edges. We derive a few further consequences from this structural result and investigate to which extent such forests of copies must be present in Ramsey graphs.Open AccessUnavoidable subgraphs in Ramsey graphs
We study subgraphs that appear in large Ramsey graphs for a given graph $F$. The recent girth Ramsey theorem of the first two authors asserts that there are Ramsey graphs such that all small subgraphs are `forests of copies of $F$' amalgamated on vertices and edges. We derive a few further consequences from this structural result and investigate to which extent such forests of copies must be present in Ramsey graphs. - Open Access
Relative Turán densities of ordered graphs
We introduce a modification of the Turán density of ordered graphs and investigate this graph parameter.
Open AccessRelative Turán densities of ordered graphs
We introduce a modification of the Turán density of ordered graphs and investigate this graph parameter.
- Open Access
Canonical Ramsey numbers for partite hypergraphs
We show that canonical Ramsey numbers for partite hypergraphs grow single exponentially for any fixed uniformity.
Open AccessCanonical Ramsey numbers for partite hypergraphs
We show that canonical Ramsey numbers for partite hypergraphs grow single exponentially for any fixed uniformity.
- Open Access
The codegree Turán density of $3$-uniform tight cycles
Given any $\varepsilon>0$ we prove that every sufficiently large $n$-vertex $3$-graph $H$ where every pair of vertices is contained in at least $(1/3+\varepsilon)n$ edges contains a copy of $C_{10}$, i.e.\ the tight cycle on $10$ vertices. In fact we obtain the same conclusion for every cycle $C_\ell$ with $\ell\geq 19$.Open AccessThe codegree Turán density of $3$-uniform tight cycles
Given any $\varepsilon>0$ we prove that every sufficiently large $n$-vertex $3$-graph $H$ where every pair of vertices is contained in at least $(1/3+\varepsilon)n$ edges contains a copy of $C_{10}$, i.e.\ the tight cycle on $10$ vertices. In fact we obtain the same conclusion for every cycle $C_\ell$ with $\ell\geq 19$. - Open Access
Turán density of cliques of order five in $3$-uniform hypergraphs with quasirandom links
We show that $3$-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than $1/3$ contain a clique on five vertices. This result is asymptotically best possible.Open AccessTurán density of cliques of order five in $3$-uniform hypergraphs with quasirandom links
We show that $3$-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than $1/3$ contain a clique on five vertices. This result is asymptotically best possible. - Open Access
On quantitative aspects of a canonisation theorem for edge-orderings
For integers $k\ge 2$ and $N\ge 2k+1$ there are $k!2^k$ canonical orderings of the edges of the complete $k$-uniform hypergraph with vertex set $[N] = \{1,2,\dots, N\}$. These are exactly the orderings with the property that any two subsets $A, B\subseteq [N]$ of the same size induce isomorphic suborderings. We study the associated canonisation problem to estimate, given $k$ and $n$, the least integer $N$ such that no matter how the $k$-subsets of $[N]$ are ordered there always exists an $n$-element set $X\subseteq [N]$ whose $k$-subsets are ordered canonically. For fixed $k$ we prove lower and upper bounds on these numbers that are $k$ times iterated exponential in a polynomial of $n$.Open AccessOn quantitative aspects of a canonisation theorem for edge-orderings
For integers $k\ge 2$ and $N\ge 2k+1$ there are $k!2^k$ canonical orderings of the edges of the complete $k$-uniform hypergraph with vertex set $[N] = \{1,2,\dots, N\}$. These are exactly the orderings with the property that any two subsets $A, B\subseteq [N]$ of the same size induce isomorphic suborderings. We study the associated canonisation problem to estimate, given $k$ and $n$, the least integer $N$ such that no matter how the $k$-subsets of $[N]$ are ordered there always exists an $n$-element set $X\subseteq [N]$ whose $k$-subsets are ordered canonically. For fixed $k$ we prove lower and upper bounds on these numbers that are $k$ times iterated exponential in a polynomial of $n$. - 1 September 2021
- Published by Cambridge University Press (CUP) in Combinatorics, Probability and Computing
- Vol. 30 (5) , 670-685
On the size-Ramsey number of grid graphs
The size-Ramsey number of a graph F is the smallest number of edges in a graph G with the Ramsey property for F, that is, with the property that any 2-colouring of the edges of G contains a monochromatic copy of F. We prove that the size-Ramsey number of the grid graph on n x n vertices is bounded from above by n(3+o(1)).
- 1 September 2021
- Published by Cambridge University Press (CUP) in Combinatorics, Probability and Computing
- Vol. 30 (5) , 670-685
On the size-Ramsey number of grid graphs
The size-Ramsey number of a graph F is the smallest number of edges in a graph G with the Ramsey property for F, that is, with the property that any 2-colouring of the edges of G contains a monochromatic copy of F. We prove that the size-Ramsey number of the grid graph on n x n vertices is bounded from above by n(3+o(1)).