Bhargav Narayanan

On the stability of the Erdős-Ko-Rado theorem

Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as an intersecting (uniform) family, this gives us a random analogue of the Erd?s-Ko-Rado theorem.

A CANONICAL RAMSEY THEOREM FOR EXACTLY m-COLOURED COMPLETE SUBGRAPHS

Given an edge colouring of a graph with a set of

Disjoint induced subgraphs of the same order and size

m

Catching a fast robber on the grid

colours, we say that the graph is (exactly)

EXACTLY m-COLOURED COMPLETE INFINITE SUBGRAPHS

m

Induced Subgraphs With Many Distinct Degrees

-coloured if each of the colours is used. We consider edge colourings of the complete graph on

Ramsey Graphs Induce Subgraphs of Many Different Sizes

\mathbb{N}

The number of hypergraphs without linear cycles

with infinitely many colours and show that either one can find an

Approximations to m-Colored Complete Infinite Hypergraphs

m

Long cycles in Hamiltonian graphs

-coloured complete subgraph for every natural number