In this talk we discuss the problems of finding large induced subgraphs of a given graph G with some degree-constraints. We survey some classical results, present some intersting and challenging open problems, and sketch solutions to some of them. 

This is based on joint works with Liam Hardiman and Michael Krivelevich. 

An independent set of size k in a finite undirected graph is a set of k vertices of the graph, no two of which are connected by an edge. The structure of independent sets of size k as k varies is of interest in probability, statistical physics, combinatorics, and computer science. In 1987, Alavi, Malde, Schwenk and Erdos conjectured that the number of independent sets of size k in a tree is a unimodal sequence (this number goes up and then it goes down), and this problem is still open. A variation on this question is: do the number of independent sets of size k form a unimodal sequence for Erdos-Renyi random graphs, or random trees? By adapting an argument of Coja-Oghlan and Efthymiou, we show unimodality for Erdos-Renyi random graphs, random bipartite graphs and random regular graphs (with high probability as the number of vertices in the graph goes to infinity, when the expected degree of a single vertex is large). The case of random trees remains open, as we can only show weak partial results there. 

A graph G on n vertices is called an alpha-expander if the external neighborhood of every vertex subset U of size |U|<=n/2 in G has size at least alpha*|U|.

Extending and improving the results of Plotkin, Rao and Smith, and of Kleinberg and Rubinfeld from the 90s, we prove that for every alpha>0, an alpha-expander G on n vertices contains every graph H with at most cn/log n vertices and edges as a minor, for c=c(alpha)>0.

Alternatively, every n-vertex graph G without sublinear separators contains all graphs with cn/logn vertices and edges as minors.

Consequently, a supercritical random graph G(n,(1+epsilon)/n) is typically minor-universal for the class of graphs with cn/log n vertices and edges.

The order of magnitude n/log n in the above results is optimal.

A joint work with Rajko Nenadov.

We present recent results on the geometry of centrally-symmetric random polytopes generated by N independent copies of a random vector X. We show that under minimal assumptions on X, for N>Cn, and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector - namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing - noise blind sparse recovery. This is joint work with the speaker's PhD student Christian Kümmerle (now at Johns Hopkins University) as well as Olivier Guédon (University of Paris-Est Marne La Vallée), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (RWTH Aachen).