We prove that for , if is an -vertex graph with chromatic number but any its proper subgraph has smaller chromatic number, then contains at most copies of cliques of size . This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai. …
I will present the short proof from that for every digraph F and every assignment of pairs of integers to its arcs, there exists an integer such that every digraph D with dichromatic number at least contains a subdivision of in which is subdivided into a directed path of …
Katona's intersection theorem states that every intersecting family satisfies , where is the shadow of . Frankl conjectured that for and every intersecting family , there is some such that , where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal …
We introduce a novel definition of orientation on the triples of a family of pairwise intersecting planar convex sets and study its properties. In particular, we compare it to other systems of orientations on triples that satisfy a natural interiority condition. Such systems, P3O (partial 3-order), are a natural generalization of posets, and include the …
The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every there exists some constant such that every -minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most vertices, to graphs that can be seen as the union of some …
An -vertex graph is called -Ramsey if it has no clique or independent set of size (i.e., if it has near-optimal Ramsey behavior). We study edge-statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a -Ramsey graph. One …
Van der Waerden's theorem states that any coloring of with a finite number of colors will contain arbitrarily long monochromatic arithmetic progressions. This motivates the definition of the van der Waerden number which is the smallest such that any -coloring of guarantees the presence of a monochromatic arithmetic progression of length …
A common technique to characterize hereditary graph classes is to exhibit their minimal obstructions. Sometimes, the set of minimal obstructions might be infinite, or too complicated to describe. For instance, for any , the set of minimal obstructions of the class of -colourable graphs is yet unknown. Nonetheless, the Roy-Gallai-Hasse-Vitaver Theorem asserts that a graph …
