Positive co-degree density of hypergraphs

The minimum positive co-degree of a non-empty $r$-graph ${H}$, denoted $\delta_{r-1}^+( {H})$, is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of ${H}$, then $S$ is contained in at least $k$ distinct hyperedges of ${H}$. Given a family ${F}$ of $r$-graphs, we introduce the {\it positive co-degree Tur\'an number} $\mathrm{co^+ex}(n, {F})$ as the maximum positive co-degree $\delta_{r-1}^+(H)$ over all $n$-vertex $r$-graphs $H$ that do not contain $F$ as a subhypergraph. In this paper we concentrate on the behavior of $\mathrm{co^+ex}(n, {F})$ for $3$-graphs $F$. In particular, we determine asymptotics and bounds for several well-known concrete $3$-graphs $F$ (e.g.\ $K_4^-$ and the Fano plane). We also show that, for $3$-graphs, the limit $$\gamma^+(F) := \limsup_{n \rightarrow \infty} \frac{\mathrm{co^+ex}(n, {F})}{n}$$ ``jumps'' from $0$ to $1/3$, i.e., it never takes on values in the interval $(0,1/3)$, and we characterize which $3$-graphs $F$ have $\gamma^+(F)=0$. Our motivation comes primarily from the study of (ordinary) co-degree Tur\'an numbers where a number of results have been proved that inspire our results

Positive co-degree Tur\'an number for C5C_5 and C5−C_5^{-}

The \emph{minimum positive co-degree} $\delta^{+}_{r-1}(H)$ of a non-empty $r$-graph $H$ is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $H$, then $S$ is contained in at least $k$ hyperedges of $H$. For any $r$-graph $F$, the \emph{positive degree Tur\'an number} $\mathrm{co}^{+}\mathrm{ex}(n,F)$ is defined as the maximum value of $\delta^{+}_{r-1}(H)$ over all $n$-vertex $F$-free non-empty $r$-graphs $H$. In this paper, we determine the positive degree Tur\'an number for $C_5$ and $C_5^{-}$.Comment: 8 page

An asymptotic bound for the strong chromatic number

The strong chromatic number $\chi_{\text{s}}(G)$ of a graph $G$ on $n$ vertices is the least number $r$ with the following property: after adding $r \lceil n/r \rceil - n$ isolated vertices to $G$ and taking the union with any collection of spanning disjoint copies of $K_r$ in the same vertex set, the resulting graph has a proper vertex-colouring with $r$ colours. We show that for every $c > 0$ and every graph $G$ on $n$ vertices with $\Delta(G) \ge cn$, $\chi_{\text{s}}(G) \leq (2 + o(1)) \Delta(G)$, which is asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu

Independence densities of hypergraphs

We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as $F$-free densities of graphs for a given graph $F.$ In the case of $k$-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real number in $[0,1].$ Finally, we extend the notion of independence density via independence polynomials

Shadow of hypergraphs under a minimum degree condition

We prove a minimum degree version of the Kruskal--Katona theorem: given $d\ge 1/4$ and a triple system $F$ on $n$ vertices with minimum degree at least $d\binom n2$, we obtain asymptotically tight lower bounds for the size of its shadow. Equivalently, for $t\ge n/2-1$, we asymptotically determine the minimum size of a graph on $n$ vertices, in which every vertex is contained in at least $\binom t2$ triangles. This can be viewed as a variant of the Rademacher--Tur\'an problem