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Register a new userAvoiding long Berge cycles, the missing cases $k=r+1$ and $k = r+2$
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The maximum size of an $r$-uniform hypergraph without a Berge cycle of length at least $k$ has been determined for all $k ge r+3$ by Furedi, Kostochka and Luo and for $k<r$ (and $k=r$, asymptotically) by Kostochka and Luo. In this paper, we settle the remaining cases: $k=r+1$ and $k=r+2$, proving a conjecture of Furedi, Kostochka and Luo.
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The areas of Ramsey theory and random graphs have been closely linked ever since ErdH{o}s famous proof in 1947 that the diagonal Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the off-diagonal Ramsey numbers $R(3,k)$. In this model, edges of $K_n$ are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted $G_{n,triangle}$. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kims celebrated result that $R(3,k) = Theta big( k^2 / log k big)$. In this paper we improve the results of both Bohman and Kim, and follow the triangle-free process all the way to its asymptotic end. In particular, we shall prove that $$ebig( G_{n,triangle} big) ,=, left( frac{1}{2sqrt{2}} + o(1) right) n^{3/2} sqrt{log n },$$ with high probability as $n to infty$. We also obtain several pseudorandom properties of $G_{n,triangle}$, and use them to bound its independence number, which gives as an immediate corollary $$R(3,k) , ge , left( frac{1}{4} - o(1) right) frac{k^2}{log k}.$$ This significantly improves Kims lower bound, and is within a factor of $4 + o(1)$ of the best known upper bound, proved by Shearer over 25 years ago.
We find a class of minimal hypersurfaces H(k) as the zero level set of Pfaffians, resp. determinants of real 2k+2 dimensional antisymmetric matrices. While H(1) and H(2) are congruent to a 6-dimensional quadratic cone resp. Hsiangs cubic su(4) invariant in R15, H(k>2) (special harmonic so(2k+2)-invariant cones of degree>3) seem to be new.
Let $phi_H^r(n)$ be the smallest integer such that, for all $r$-graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $phi_H^r(n)$ parts, of which every part either is a single edge or forms an $r$-graph isomorphic to $H$. The function $phi^2_H(n)$ has been well studied in literature, but for the case $rge 3$, the problem that determining the value of $phi_H^r(n)$ is widely open. Sousa (2010) gave an asymptotic value of $phi_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, and determined the exact value of $phi_H^r(n)$ in some special cases. In this paper, we first give the exact value of $phi_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, which improves Sousas result. Second we determine the exact value of $phi_H^r(n)$ when $H$ is an $r$-graph consisting of exactly $k$ independent edges.
Let $G=(V,E)$ and $H$ be two graphs. Packing problem is to find in $G$ the largest number of independent subgraphs each of which is isomorphic to $H$. Let $Usubset{V}$. If the graph $G-U$ has no subgraph isomorphic to $H$, $U$ is a cover of $G$. Covering problem is to find the smallest set $U$. The vertex-disjoint tree packing was not sufficiently discussed in literature but has its applications in data encryption and in communication networks such as multi-cast routing protocol design. In this paper, we give the kind of $(k+1)$-connected graph $G$ into which we can pack independently the subgraphs that are each isomorphic to the $(2^{k+1}-1)$-order perfect binary tree $T_k$. We prove that in $G$ the largest number of vertex-disjoint subgraphs isomorphic to $T_k$ is equal to the smallest number of vertices that cover all subgraphs isomorphic to $T_k$. Then, we propose that $T_k$ does not have the emph{ErdH{o}s-P{o}sa} property. We also prove that the $T_k$ packing problem in an arbitrary graph is NP-hard, and propose the distributed approximation algorithms.
The ErdH{o}s-Simonovits stability theorem states that for all epsilon >0 there exists alpha >0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - alpha n^2, then one can remove epsilon n^2 edges from G to obtain an r-partite graph. Furedi gave a short proof that one can choose alpha=epsilon. We give a bound for the relationship of alpha and varepsilon which is asymptotically sharp as epsilon to 0.
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