Draw a graph on $10$ vertices with no more than $20$ edges that contains no independent set of size $3$.
So I was trying to draw the above graph but I was kind of stuck. What I basically did was draw a bipartite graph with $5$ vertices on each side and then $20$ edges randomly connecting between them. I suppose it was similar to $K5, 5$, though I don't believe it is quite the same. I was also a little confused on how to identify an independent set in a graph so it was hard for me to verify my graph.
If anyone knows how to approach this problem, some assistance will be highly appreciated!
Take $G=K_5+ K_5$ (the disjoint union of two 5-cliques). Then $G$ has $2\cdot 5=10$ vertices and $2\cdot \binom{5}{2}=20$ edges and any independent set of $G$ has cardinality less than $3$ because it has no more than one vertex from each $K_5$.
P.S. I don't think that by selecting $20$ edges in $K_{5,5}$ we can obtain a graph $G$ that contains no independent set of size $3$. You should be able to get a contradiction by noting that the average degree in $G$ is $2\cdot 20/10=4$.