Let $P(n)$ be: for every planar graph $G = (V,E)$ with $|V| = n$ there is a partition of V to $V = V_1 \cup V_2 \cup V_3$ such that $V_1 \cap V_2, V_1 \cap V_3, V_2 \cap V_3 = \emptyset$, where for all $1 \leq i \leq 3$ the graph that is formed by $V_i$ is acyclic.
We call good partition the partitions satisfying the property.
$P(3)$ is trivial. $\{v_1\},\{v_2\},\{v_3\}$ is a good partition.
Suppose that $P(n)$ is true and let us show that $P(n+1)$ hold.
$V=\{v_1,\dots,v_{n+1}\}$, let $\bar{V}^i$ be the set $\{v_1,\dots,v_{i-1},v_{i+1},\dots,v_{n+1}\}$ by hypothesis there is a good partition of $\bar{V}^i$.
Three cases:
there exists $i$ such that there exists a good partition of $\bar{V}^i$ such that $\forall (v_{i},v)\in E, v\notin V_1$ then $V_1\cup\{v_{i}\},V_2,V_3$ is a partition of $V$ satifying $P(n+1)$.
or, there exists $i$ such that there exists a good partition of $\bar{V}^i$ such that $\exists v\in V_1,(v_{i},v)\in E$ and $\forall (v_{i},v')\in E,v\neq v'\implies v'\notin V_1$ then $V_1\cup\{v_{i}\},V_2,V_3$ is a partition of $V$ satifying $P(n+1)$.
Or, for all $i$ and for all good partitions of $\bar{V}^i$ there are at least two neighbour of $v_{i}$ in each set $V_j$. Hence the degree of each vertices is at least 6. But: see last corollary of this page : "Every finite, simple, planar graph has a vertex of degree less than 6" contradiction with "G" is planar.