Lower Bounds for the Number of Edges in Hypergraphs of Certain Classes

The classical Erdös-Hajnal extremal problem of hypergraph coloring and some of its generalizations are studied and a new lower asymptotic bound is found. Inequality with bounds are compared and Erdös lower bound and bound for all valued greater than or equal to 3 are improved. An n-uniform hypergraph is considered and an improvement of Erdös and Lovász theorem is proposed. A hypergraph is said to be simple if any two different edges have at most one common vertex. The best asymptotic lower bounds are obtained asserting the existence of a positive integer satisfying an equality. Bounds obtained are found to asymptotically improve the other bounds in a certain domain.