In summer 2021, I'm doing a research in graph theory and combinatorics, supervised by Dr. Aled Walker. The preprint can be found here and the abstract of the result is as below:

Throughout this summer research placement, we have investigated the size of the number of induced subgraphs of bipartite graphs. Given a bipartite graph with $m$ edges, we proved that the size of induced subgraph has $\Omega(m/(\log m)^{10})$ edges, optimizing the results from Narayanan, Sahasrabudhe and Tomon. We also introduce the concept of $C$-$bipartite$-$Ramsey$, proving that in `most' cases, these graphs have multiplication tables of $\Omega(e(G))$ in size, which gives evidence to the conjecture that $K_{n,n}$ is the minimiser of the multiplication table on $n^2$ edges. While reading "Ramsey graphs induce subgraphs of quadratically many sizes" by Kwan and Sudakov, we found a small non-trivial mistake in it and we successfully fixed it afterwards. We also tried to tackle the conjecture that given $m$ edges, one can always find a bipartite graph with a multiplication table of size $o(m)$ as $m \rightarrow \infty$.