How to get even numbers from adding pairs? (best title I thought of) if I want to make a subset of numbers for which sums of pairs involve all even numbers what is the subset with the smallest density?
for example {1 , 3 , 5} can pair to get 2 :1+1 , 4 :1+3 , 6 :3+3, 8: 5+3, 10 :5+5 so all even numbers until 10 can be achieved with a set of 3 numbers.
I made a java program that gets the smallest possible set by trying all possibilities. It reached 42 then (out of memory error) but the size of the set seemed to be approximately :
Size =The Largest Even Number/ log2(The Largest Even Number)
for 42 it is 7 , for 22 it is 5. Does this hold for larger numbers? And is there a theorem for that?
 A: For fixed $n$, you can solve the problem via integer linear programming as follows.  For $i\in\{1,\dots,n\}$, let binary decision variable $x_i$ indicate whether element $i$ is selected.  The problem is to minimize $\sum_i x_i$ subject to
$$\sum_{i\le k-i} x_i x_{k-i} \ge 1 \quad \text{for $k\in\{2,4,\dots,n\}$} \tag1\label1.$$
You can linearize these quadratic constraints \eqref{1} by introducing (nonnegative or binary) variables $y_{ij}$ to represent the products $x_i x_j$, together with linear constraints
\begin{align}
\sum_{i\le k-i} y_{i,k-i} &\ge 1 &&\text{for $k\in\{2,4,\dots,n\}$} \tag2\label2 \\
y_{ij} &\le x_i && \text{for $1\le i \le j \le n$} \tag3\label3 \\
y_{ij} &\le x_j && \text{for $1\le i \le j \le n$} \tag4\label4
\end{align}
The optimal objective values for even $n$ up to $100$ are:
\begin{matrix}
n & \sum_i x_i \\
\hline
2  &  1  \\
4  &  2  \\
6  &  2  \\
8  &  3  \\
10 &  3  \\
12 &  4  \\
14 &  4  \\
16 &  4  \\
18 &  4  \\
20 &  5  \\
22 &  5  \\
24 &  5  \\
26 &  5  \\
28 &  6  \\
30 &  6  \\
32 &  6  \\
34 &  6  \\
36 &  7  \\
38 &  7  \\
40 &  7  \\
42 &  7  \\
44 &  8  \\
46 &  8  \\
48 &  8  \\
50 &  8  \\
52 &  8  \\
54 &  8  \\
56 &  9  \\
58 &  9  \\
60 &  9  \\
62 &  9  \\
64 &  9  \\
66 &  9  \\
68 &  10 \\
70 &  10 \\
72 &  10 \\
74 &  10 \\
76 &  10 \\
78 &  10 \\
80 &  10 \\
82 &  10 \\
84 &  11 \\
86 &  11 \\
88 &  11 \\
90 &  11 \\
92 &  11 \\
94 &  11 \\
96 &  12 \\
98 &  12 \\
100 &  12
\end{matrix}
