Topological cliques in sparse expanders

In the paper, we focus on embedding clique immersions and subdivisions within sparse expanders, and we derive the following main results: (1) For any $0< \eta< 1/2$, there exists $K>0$ such that for sufficiently large $n$, every $(n,d,\lambda)$-graph $G$ contains a $K_{(1-5\eta)d}$-immersion when $d\geq K\lambda$. (2) For any $\varepsilon>0$ and $0<\eta <1/2$, the following holds for sufficiently large $n$. Every $(n,d,\lambda)$-graph $G$ with $2048\lambda/\eta^2<d\leq \eta n^{1/2-\varepsilon}$ contains a $K_{(1-\eta)d}^{(\ell)}$-subdivision, where $\ell = 2 \left\lceil \log(\eta^2n/4096)\right\rceil + 5$. (3) There exists $c>0$ such that the following holds for sufficiently large $d$. If $G$ is an $n$-vertex graph with average degree $d(G)\geq d$, then $G$ contains a $K_{c d}^{(\ell)}$-immersion for some $\ell\in \mathbb{N}$. In 2018, Dvo{\v{r}}{\'a}k and Yepremyan asked whether every graph $G$ with $\delta(G)\geq t$ contains a $K_t$-immersion. Our first result shows that it is asymptotically true for $(n,d,\lambda)$-graphs when $\lambda=o(d)$. In addition, our second result extends a result of Dragani{\'c}, Krivelevich and Nenadov on balanced subdivisions. The last result generalises a result of DeVos, Dvo{\v{r}}{\'a}k, Fox, McDonald, Mohar, Scheide on $1$-immersions of large cliques in dense graphs.