The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of Hàn, Person, and Schacht [SIAM J. Disc. Math., 23 (2009), pp. 732-748] who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an n-vertex 3-graph isIn this paper, we improve on this result, giving a family of degree sequence results, all of which imply the result of Hàn, Person and Schacht and additionally allow one-third of the vertices to have degree 1 9 n 2 below this threshold. Furthermore, we show that this result is, in some sense, tight.
The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of Hàn, Person and Schacht [6] who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an n-vertex 3-graph is 5 9 + o(1) n 2 . In this paper we improve on this result, giving a family of degree sequence results, all of which imply the result of Hàn, Person and Schacht, and additionally allow one third of the vertices to have degree 1 9 n 2 below this threshold. Furthermore, we show that this result is, in some sense, tight.
For integers r and n, where n is sufficiently large, and for every set X ⊆ [n] we determine the maximal left-compressed intersecting families A ⊆ [n] r which achieve maximum hitting with X (i.e. have the most members which intersect X). This answers a question of Barber, who extended previous results by Borg to characterise those sets X for which maximum hitting is achieved by the star.
For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and Samotij proved an upper bound on the threshold function for the property that $G(n,p) \to (H_1,\dots,H_r)$, thereby resolving the $1$-statement of the Kohayakawa--Kreuter conjecture. %We show that to prove the $0$-statement it suffices to prove a deterministic colouring result, which says that if $G$ is not too dense then $G \not \to (H_1,\dots,H_r)$. We extend upon the many partial results for the $0$-statement, by resolving it for a large number of cases, which in particular includes (but is not limited to) when $r \geq 3$, when $H_2$ is strictly $2$-balanced and not bipartite, or when $H_1$ and $H_2$ have the same $2$-densities.
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