Let $r$ be any positive integer. We prove that for every sufficiently large $k$ there exists a $k$-chromatic vertex-critical graph $G$ such that $\chi (G-R)=k$ for every set $R \subseteq E(G)$ with $|R|\le r$. This partially solves a problem posed by Erdős in 1985, who asked whether the above statement holds for $k \ge 4$.

Daisies are a special type of hypergraph introduced by Bollobás, Leader and Malvenuto. An $r$-daisy determined by a pair of disjoint sets $K$ and $M$ is the $(r+|K|)$-uniform hypergraph $\{K\cup P\,{:}\, P\in M^{(r)}\}$. Bollobás, Leader and Malvenuto initiated the study of Turán type density problems for daisies. This paper deals with Ramsey numbers of daisies, which are natural generalisations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of $r$-daisies and also for special cases where the size of the kernel is bounded.

A $(k+r)$-uniform hypergraph $H$ on $(k+m)$ vertices is an $(r,m,k)$-daisy if there exists a partition of the vertices $V(H)=K\cup M$ with $|K|=k$, $|M|=m$ such that the set of edges of $H$ is all the $(k+r)$-tuples $K\cup P$, where $P$ is an $r$-tuple of $M$. We obtain an $(r-2)$-iterated exponential lower bound to the Ramsey number of an $(r,m,k)$-daisy for $2$-colours. This matches the order of magnitude of the best lower bounds for the Ramsey number of a complete $r$-graph.

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least

\begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*}

$\exp\!(n\log n-\Theta (n))$

$\ell$

$(k-\ell )\mid n$

$(k-\ell )\mid k$

$(k-\ell )\nmid k$

$\ell \lt k/2$

A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$-colouring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$. We prove a result which says that if one replaces $K_n^k$ in Gyárfás’ theorem by any ‘expansive’ $k$-uniform hypergraph on $n$ vertices (that is, a $k$-uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.

Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$-partite $r$-uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$, there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$, then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$). We prove our results in this dual setting.

A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every fixed $C$, if each vertex $v \in \bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$, then the (list) chromatic number of $\bigcup _{i=1}^m G_i$ is at most $D + o(D)$. This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy and Postle, and we derive this result from a more general list colouring result in the setting of ‘colour degrees’ that also implies a result of Reed and Sudakov.

Given $\alpha \gt 0$ and an integer $\ell \geq 5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $\alpha n$ edges contains a copy of $C_\ell ^{-}$, a tight cycle on $\ell$ vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidický.

Alweiss, Lovett, Wu, and Zhang introduced $q$-spread hypergraphs in their breakthrough work regarding the sunflower conjecture, and since then $q$-spread hypergraphs have been used to give short proofs of several outstanding problems in probabilistic combinatorics. A variant of $q$-spread hypergraphs was implicitly used by Kahn, Narayanan, and Park to determine the threshold for when a square of a Hamiltonian cycle appears in the random graph $G_{n,p}$. In this paper, we give a common generalization of the original notion of $q$-spread hypergraphs and the variant used by Kahn, Narayanan, and Park.

We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$-uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle.

In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$-uniform hypergraphs by proving a $3$-uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.

A hypergraph $\mathcal{F}$ is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of $\mathcal{F}$ . Mubayi and Verstraëte showed that for every $k \ge d+1 \ge 3$ and $n \ge (d+1)k/d$ every $k$ -graph $\mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $\binom{n-1}{k-1}$ edges. They conjectured that the same conclusion holds for all $d \ge k \ge 4$ and sufficiently large $n$ . We confirm their conjecture by proving a stronger statement.

They also conjectured that for $m \ge 4$ and sufficiently large $n$ the maximum size of a $3$ -graph on $n$ vertices without a non-trivial intersecting subgraph of size $3m+1$ is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.

To model the dynamics of discrete deterministic systems, we extend the Petri netsframework by a priority relation between conflicting transitions, which is encoded byorienting the edges of a transition conflict graph. The aim of this paper is to gain someinsight into the structure of this conflict graph and to characterize a class of suitableorientations by an analysis in the context of hypergraph theory.

A hypergraph is Helly if every family of hyperedges of it, formed by pairwiseintersecting hyperedges, has a common vertex. We consider the concepts ofbipartite-conformal and (colored) bipartite-Helly hypergraphs. In the same way asconformal hypergraphs and Helly hypergraphs are dual concepts, bipartite-conformal andbipartite-Helly hypergraphs are also dual. They are useful for characterizing bicliquematrices and biclique graphs, that is, the incident biclique-vertex incidence matrix andthe intersection graphs of the maximal bicliques of a graph, respectively. These conceptsplay a similar role for the bicliques of a graph, as do clique matrices and clique graphs,for the cliques of the graph. We describe polynomial time algorithms for recognizingbipartite-conformal and bipartite-Helly hypergraphs as well as biclique matrices.


We deal here with a scheduling problem GPPCSP (Generalized Parallelism and Preemption Constrained Scheduling Problem) which is an extension of both the well-known Resource Constrained Scheduling Problem and the Scheduling Problem with Disjunctive Constraints. We first propose a reformulation of GPPCSP: according to it, solving GPPCSP means finding a vertex of the Feasible Vertex Subset of an Antichain Polyhedron. Next, we state several theoretical results related to this reformulation process and to structural properties of this specific Feasible Vertex Subset (connectivity, ...). We end by focusing on the preemptive case of GPPCSP and by identifying specific instances of GPPCSP which are such that any vertex of the related Antichain Polyhedron may be projected on its related Feasible Vertex Subset without any deterioration of the makespan. For such an instance, the GPPCSP problem may be solved in a simple way through linear programming.