An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs H and G , the anti-Ramsey number a r ( ) in maximum colors coloring so that there does not exist copy . Li et al. determined k -matchings complete bipartite graphs. Jin Zang showed uniqueness extremal coloring. In this paper, as generalization these results, we determine K n 1 … M -partite -un...

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability of Erd\H{o}s and Simonovits shows if a with $n$ vertices has close to the maximal $t_r(n)$ edges, then it being In this paper we determine exactly graphs at least $m$ edges are farthest from $r$-partite, for any $m\ge t_r(n) - \delta_r n^2$. This extends work by Erd\H{o}s, Gy\H{o}ri Simonovits, proves con...

An ordered r-graph is an r-uniform hypergraph whose vertex set linearly ordered. Given 2?k?r, H interval k-partite if there exist at least k disjoint intervals in the ordering such that every edge of has nonempty intersection with each and contained their union. Our main result implies ?>k?1, then for d>0 n-vertex dn? edges some m?n m-vertex subgraph ?(dm?) edges. This extension to r-graphs obs...

Abstract The book graph $B_n ^{(k)}$ consists of $n$ copies $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ Here consider natural off-diagonal variant $r(B_{cn} B_n^{(k)})$ for fixed $c \in (0,1]$ more general setting, show that an interesting dichotomy emerges: very small $c$ , simple $k$ -partite construction ...

We determine the strong total chromatic number of the complete h-uniform hypergraph Kh, and the complete h-partite hypergraph K,

Identifying the $k$-partite entanglement and $k$-nonseparability of general $N$-partite quantum states are fundamental issues in information theory. By use computable inequalities nonlinear operators, we present some simple powerful criteria that works very well allow for a inexpensive test whole hierarchy $k$-separability systems with $k$ running from $N$ down to 2. We illustrate their strengt...