Related papers: The Lemmens-Seidel conjecture and forbidden subgra…
We show that for any natural number $s$, there is a constant $\gamma$ and a subgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size…
An antimagic labeling of a graph $G$ with $m$ edges is a bijection from $E(G)$ to $\{1,2,\ldots,m\}$ such that for all vertices $u$ and $v$, the sum of labels on edges incident to $u$ differs from that for edges incident to $v$. Hartsfield…
We show that for any natural number $k \ge 1$, any oriented graph $D$ of minimum semidegree at least $(3k- 2)/4$ contains an antidirected path of length $k$. In fact, a slightly weaker condition on the semidegree sequence of $D$ suffices,…
We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we…
An old and well-known conjecture of Frankl and F\"{u}redi states that the Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite…
In 2011, the second author conjectured that every line graph $G$ satisfies $\chi(G)\le \max\{\omega(G),\frac{5\Delta(G)+8}{6}\}$. This conjecture is best possible, as shown by replacing each edge in a 5-cycle by $k$ parallel edges, and…
A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs).…
The Komlos conjecture in discrepancy theory states that for some constant K and for any m by n matrix A whose columns lie in the unit ball there exists a +/- 1 vector x such that the infinity norm of Ax is bounded above by K. This…
We prove Ahlswede- Khachatrian conjecture. From this conjecture follows of several other conjectures including Manickam-Mikl\'{o}s-Singhi conjecture.
In 1989 H. Tverberg proposed a quite general conjecture in Discrete geometry, which could be considered as the common basis for many results in Combinatorial geometry and at the same time as a discrete analogue of the common transversal…
The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…
In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$…
In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The ``small quasi-kernel conjecture,'' proposed by Erd\H{o}s and Sz\'ekely…
We show that diagrammatic sets, a topologically sound alternative to polygraphs and strict $\omega$-categories, admit an internal notion of equivalence in the sense of coinductive weak invertibility. We prove that equivalences have the…
In 1972, Kainen proved a general lower bound on the crossing number of a graph in a closed surface and conjectured that this bound is tight when the graph is either a complete graph or a complete bipartite graph, and the surface is of genus…
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…
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…
A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…
In `A survey of two-graphs' \cite{Sei}, J.J. Seidel lays out the connections between simple graphs, two-graphs, equiangular lines and strongly regular graph. It is well known that there is a one-to-one correspondence between regular…
A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the…