Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed find asymptotic results for very large graphs, so it seems that the not suitable finding numbers. But this intuition wrong, and we will develop technique do just in paper. We new upper bounds many graph hypergraph As result, prove values $R(K_4^-,K_4^-,K_4^-)=28$, $R(...

For a graph $G$, let $P(G,lambda)$ denote the chromatic polynomial of $G$. Two graphs $G$ and $H$ are chromatically equivalent if they share the same chromatic polynomial. A graph $G$ is chromatically unique if any graph chromatically equivalent to $G$ is isomorphic to $G$. A $K_4$-homeomorph is a subdivision of the complete graph $K_4$. In this paper, we determine a family of chromatically uni...

In this paper, it is proved that all simple $K_4$-groups of type $L_2(q)$ can be characterized by their maximum element orders together with their orders. Furthermore, the automorphism groups of simple $K_4$-groups of type $L_2(q)$ are also considered.

for a graph $g$, let $p(g,lambda)$ denote the chromatic polynomial of $g$. two graphs $g$ and $h$ are chromatically equivalent if they share the same chromatic polynomial. a graph $g$ is chromatically unique if any graph chromatically equivalent to $g$ is isomorphic to $g$. a $k_4$-homeomorph is a subdivision of the complete graph $k_4$. in this paper, we determine a family of chromatically uni...

Let $\mathbf L_k$ be the holomorphic line bundle of degree $k \in \mathbb Z$ on projective line. Here, tuples $(k_1 k_2 k_3 k_4)$ for which there does not exists homogeneous non-split supermanifolds $CP^{1|4}_{k_1 k_4}$ associated with vector L_{−k_1} \oplus \mathbf L _{−k_2} L_{−k_3} L_{−k_4}$ are classified. \\For many types remaining tuples, listed cocycles that determine supermanifolds. \\P...

in this paper, it is proved that all simple $k_4$-groups of type $l_2(q)$ can be characterized by their maximum element orders together with their orders. furthermore, the automorphism groups of simple $k_4$-groups of type $l_2(q)$ are also considered.