Erdős Problem #548: the Erdős–Sós conjecture

For every $n,k$ with $k+1\le n$, every simple graph $G$ on $n$ vertices satisfying $$|E(G)|\ge \frac{k-1}{2}n+1$$ contains every tree on $k+1$ vertices. The proof counts pairs $(\pi,j)$ where $\pi=(v_1,\ldots,v_n)$ is an ordering of the host vertices and $v_1v_j$ is an edge. There are exactly $$2|E(G)|(n-1)!$$ such pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus $$(k-1)n!.$$ If the target tree is absent, the rooted-copy term vanishes and one obtains $$2|E(G)|\le (k-1)n,$$ contradicting the density hypothesis.

For every $n,k$ with $k+1\le n$, every simple graph $G$ on $n$ vertices satisfying $$|E(G)|\ge \frac{k-1}{2}n+1$$ contains every tree on $k+1$ vertices. The proof counts pairs $(\pi,j)$ where $\pi=(v_1,\ldots,v_n)$ is an ordering of the host vertices and $v_1v_j$ is an edge. There are exactly $$2|E(G)|(n-1)!$$ such pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus $$(k-1)n!.$$ If the target tree is absent, the rooted-copy term vanishes and one obtains $$2|E(G)|\le (k-1)n,$$ contradicting the density hypothesis.