In my introductory course on graph theory the following statement was proven.
Any finite graph $G$ with at least one edge contains an induced subgraph $H$ such that $\delta(H) > \frac{d(H)}{2}\geq \frac{d(G)}{2}$.
Informally, the theorem claims the existence of an induced subgraph $H$ that is denser than or equally dense to $G$, and where the “density is relatively uniformly distributed.” My lecturer mentioned that sometimes the theorem, as well as one of its proofs (one proof gives an algorithm to construct such $H$) is useful. Can you reference any proofs where the theorem is used? Do you know any applications of it?
If we have a graph $G$ with minimum degree $k$, then we can find any tree $T$ on $k+1$ vertices in it, by the greedy algorithm.
(Briefly: order the vertices of $T$ as $v_0, v_1, \dots, v_k$ so that each $v_i$ has one neighbor among $\{v_0, v_1, \dots, v_{i-1}\}$. Then find an edge-preserving $f : V(T) \to V(G)$ step by step, choosing $f(v_0), f(v_1), \dots$ one at a time. Choosing $f(v_0)$ is arbitrary. When we get to $v_i$, let $v_j$ be its neighbor in $T$. We want $f(v_i)$ to be a neighbor of $f(v_j)$ distinct from all the other vertices $f(v_0), \dots, f(v_{i-1})$. That eliminates $i-1 \le k-1$ of the neighbors of $f(v_j)$, but $f(v_j)$ has at least $k$ neighbors, so we can still do this.)
Using the average-to-minimum-degree lemma, we can say that an $n$-vertex graph that does not contain the tree $T$ can have at most $(k-1)n$ edges. If it has $(k-1)n + 1$ edges, its average degree is $2(k-1) + \frac 2n$, so it has a subgraph with minimum degree at least $k-1 + \frac1n$; since $k$ is an integer, the subgraph has minimum degree at least $k$, and then we use the result above.
I think this is still the best result that's known. The Erdős-Sós conjecture, however, says that the true maximum is $\frac{k-1}{2} \cdot n$ edges (which is achieved, for example, by a union of $\frac nk$ $k$-cliques). We know that this is true for many trees, but not all trees.