The bipartite $K_{2,2}$-free process and bipartite Ramsey number $b(2, t)$
Abstract
The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such that every blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or a red $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an empty graph on vertex set $X\cup Y$, $|X|=|Y|=n$. At each step, a random edge from $X\times Y$ is added under the restriction that no $K_{2,2}$ is formed. This step is repeated until no more edges can be added. In this note, we analyze this process and show that the resulting graph witnesses that $b(2,t) =\Omega \left(t^{3/2}/\log t \right)$, thereby improving the best known lower bound.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2018
- DOI:
- 10.48550/arXiv.1808.02139
- arXiv:
- arXiv:1808.02139
- Bibcode:
- 2018arXiv180802139B
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 12 pages