I am studying about Ramsey theorem and I came across this question as mentioned below: Suppose $R(K_3; k)$ be such minimal n that in every edge colouring of $K_n$ with k colors there is a monochromatic $K_3$ (that is such that all edges have the same color.). So how do I show that $R(K_3; k) > 2^k$
Can anyone please help me with this and thanks in advance.
This is obviously true for $k=0$. Suppose you can edge-color $K_{2^k}$ with $k$ colors so that there is no monochromatic $K_3$. Partition vertices of $K_{2^{k+1}}$ into $2$ sets $U$ and $V$ of size $2^k$ and edge-color the two induced subgraphs $K_{2^k}$ on $U$ and $V$ with colors $1$ through $k$, then color all edges between a vertex in $U$ and a vertex in $V$ with color $k+1$. Check that this coloring has no monochromatic $K_3$.
To find an edge coloring without a monochromatic $K_3$ non-inductively, label the vertices of $K_{2^k}$ with binary strings of length $k$, then color an edge with color $i\in[1,k]$ if the leftmost bit in which the two endpoints' labels differ is the $i$th bit.