Tapas Kumar Mishra

Lower Bounds for Ramsey Numbers for Complete Bipartite and 3-Uniform Tripartite Subgraphs

In this paper we establish lower bounds of the number R′(a,b) so that any bicoloring of the edges of the complete undirected graph K n with n ≥ R′(a,b) vertices, always admits a monochromatic complete bipartite subgraph K a,b , where a and b are natural numbers. We show that R′(2,b) > 2b + 1 for b ≥ 4. We establish a lower bound for R′(a,b) using the probabilistic method that improves the lower bound given by Chung and Graham [4]. Further, we also use Lovasz’ local lemma to derive a better lower bound for R′(a,b). We define R′(a,b,c) be the minimum number n such that any n-vertex 3-uniform hypergraph G(V,E), or its complement G′(V,E c ) contains a K a,b,c . Here, K a,b,c is defined as the complete tripartite 3-uniform hypergraph with vertex set A ∪ B ∪ C, where the A, B and C have a, b and c vertices respectively, and K a,b,c has abc 3-uniform hyperedges {u,v,w}, u ∈ A, v ∈ B and w ∈ C. We derive lower bounds for R′(a,b,c) using probabilistic methods.

Lower bounds for Ramsey numbers for complete bipartite and 3-uniform tripartite subgraphs

Let R(Ka,b, Kc,d) be the minimum number n so that any n-vertex simple undirected graph G contains a Ka,b or its complement G′ contains a Kc,d. We demonstrate constructions showing that R(K2,b, K2,d) > b + d + 1 for d ≥ b ≥ 2. We establish lower bounds for R(Ka,b, Ka,b) and R(Ka,b, Kc,d) using probabilistic methods. We define R′(a, b, c) to be the minimum number n such that any n-vertex 3-uniform hypergraph G(V, E), or its complement G′(V, E) contains a Ka,b,c. Here, Ka,b,c is defined as the complete tripartite 3-uniform hypergraph with vertex set A∪B∪C, where the A, B and C have a, b and c vertices respectively, and Ka,b,c has abc 3-uniform hyperedges {u, v, w}, u ∈ A, v ∈ B and w ∈ C. We derive lower bounds for R′(a, b, c) using probabilistic methods. We show that R′(1, 1, b) ≤ 2b + 1. We have also generated examples to show that R′(1, 1, 3) ≥ 6 and R′(1, 1, 4) ≥ 7.

Induced-bisecting families of bicolorings for hypergraphs

Abstract Two n -dimensional vectors A and B , A , B ∈ R n , are said to be trivially orthogonal if in every coordinate i ∈ [ n ] , at least one of A ( i ) or B ( i ) is zero. Given the n -dimensional Hamming cube { 0 , 1 } n , we study the minimum cardinality of a set V of n -dimensional { − 1 , 0 , 1 } vectors, each containing exactly d non-zero entries, such that every ‘possible’ point A ∈ { 0 , 1 } n in the Hamming cube has some V ∈ V which is orthogonal, but not trivially orthogonal, to A . We give asymptotically tight lower and (constructive) upper bounds for such a set V except for the case where d ∈ Ω ( n 0 . 5 + ϵ ) and d is even, for any ϵ , 0 ϵ ≤ 0 . 5 .