$k\ge20$ integer.
Is there a graph that has group nodes $V={v_1,v_2,\ldots,v_k}$ so that the degree of node $v_i$ is ${2k+1\choose i}$ for $1\le i\le k$. If No-prove. If Yes-give an example. thanks!
*I forgot to mention that the graph maybe has multiple edges
Assuming you allow edges in parallel and/or loops, consider that:
$$\sum_{i=0}^{k} \binom{2k+1}{i} = \sum_{i=0}^{k} \binom{2k+1}{2k+1 -i} = \sum_{i=k+1}^{2k+1} \binom{2k+1}{i}$$
But we also know:
$$\sum_{i=0}^{2k+1} \binom{2k+1}{i} = \sum_{i=0}^{k} \binom{2k+1}{i} + \sum_{i=k+1}^{2k+1} \binom{2k+1}{i} = 2^{2k+1}$$
So you have:
$$\sum_{i=0}^{k} \binom{2k+1}{i} = \frac{2^{2k+1}}{2}$$
And thus:
$$\sum_{i=1}^{k} \binom{2k+1}{i} = 2^{2k}-1$$
What does this tell you about the graph? See the Handshaking Lemma.