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  1. Abstract

    In this paper, we consider a randomized greedy algorithm for independent sets inr‐uniformd‐regular hypergraphsGonnvertices with girthg. By analyzing the expected size of the independent sets generated by this algorithm, we show that, whereconverges to 0 asg → for fixeddandr, andf(d, r) is determined by a differential equation. This extends earlier results of Garmarnik and Goldberg for graphs [8]. We also prove that when applying this algorithm to uniform linear hypergraphs with bounded degree, the size of the independent sets generated by this algorithm concentrate around the mean asymptotically almost surely.

     
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  2. Abstract An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braß–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting. 
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  3. The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for $n \geqslant k \geqslant t \geqslant 2$, we consider a collection of $k$ families $\mathcal{A}_i: 1 \leq i \leqslant k$ where $\mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \}$ so that $A_{1, i_1} \cap \cdots \cap A_{k,i_k} \neq \varnothing$ if and only if there are at least $t$ distinct indices $i_1,i_2,\dots,i_k$. Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size $\beta_{k,t}(n)$ of the families with ground set $[n]$. 
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  4. Abstract Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n -vertex graph of minimum degree at least d , then $$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$ In this paper the main result is that if G is any n -vertex d -regular graph of girth at least five, then $$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$ for some constant c independent of d . This result is sharp in the sense that as $d \rightarrow \infty$ , almost all d -regular n -vertex graphs G of girth at least five have $$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$ Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs $K_{d,d}$ , then ${\gamma_\circ}(G) = \frac{n}{2}$ . We also prove that there are n -vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that ${\gamma_\circ}(G) \sim {n}/{2}$ as $d \rightarrow \infty$ . Therefore both the girth and regularity conditions are required for the main result. 
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