Related papers: A hypergraph blow-up lemma
A result of Balogh, Csaba, Jing and Pluh\'ar yields the minimum degree threshold that ensures a $2$-coloured graph contains a perfect matching of significant colour-bias (i.e., a perfect matching that contains significantly more than half…
This work studies the typical structure of sparse $H$-free graphs, that is, graphs that do not contain a subgraph isomorphic to a given graph $H$. Extending the seminal result of Osthus, Pr\"omel, and Taraz that addressed the case where $H$…
The shrinking operation converts a hypergraph into a graph by choosing, from each hyperedge, two endvertices of a corresponding graph edge. A hypertree is a hypergraph which can be shrunk to a tree on the same vertex set. Klimo\v{s}ov\'{a}…
We conjecture that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of the combined degree, provided it grows as the number of vertices…
The hypergraph regularity lemma -- the extension of Szemer\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several…
Given a graph $H$, a balanced subdivision of $H$ is a graph obtained from $H$ by subdividing every edge the same number of times. In 1984, Thomassen conjectured that for each integer $k\ge 1$, high average degree is sufficient to guarantee…
One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two.…
We asymptotically determine the maximum density of subgraphs isomorphic to $H$, where $H$ is any graph containing a dominating vertex, in graphs $G$ on $n$ vertices with bounded maximum degree and bounded clique number. That is, we…
We define a new combinatorial object, which we call a labeled hypergraph, uniquely associated to any square-free monomial ideal. We prove several upper bounds on the regularity of a square-free monomial ideal in terms of simple…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide…
Hansel's lemma states that $\sum_{H\in \mathcal{H}}|H| \geq n \log_2 n$ holds where $\mathcal{H}$ is a collection of bipartite graphs covering all the edges of $K_n$. We generalize this lemma to the corresponding multigraph covering problem…
This MSci thesis surveys results in extremal graph theory, in particular relating to Hamilton cycles. Szem\'eredi's Regularity Lemma plays a central role. We also investigate the robust outexpansion property for digraphs. Kelly showed that…
The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence…
We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph…
In line with the recent development in topological graph theory, we are considering undirected graphs that are allowed to contain {\em multiple edges}, {\em loops}, and {\em semi-edges}. A graph is called {\em simple} if it contains no…
The Lagrangian density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all $F$-free $r$-uniform hypergraphs. For an $r$-graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge…
In 1987, Kolaitis, Pr\"omel and Rothschild proved that, for every fixed $r \in \mathbb{N}$, almost every $n$-vertex $K_{r+1}$-free graph is $r$-partite. In this paper we extend this result to all functions $r = r(n)$ with $r \leqslant (\log…
The colored neighborhood metric for sparse graphs was introduced by Bollob\'as and Riordan. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm. We prove that even in this refined sense, the…
Szemer\'edi's regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lov\'asz and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst, Geometric…
We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension…