Prove that an $s$ element subset of $1,2,...,n$ must have two distinct subsets with the same sum.

Question

A number of problems on math.stackexchange have taken the form

Prove that an $s$ element subset of $1,2,...,n$ must have two distinct subsets with the same sum.

(For example discrete math about Pigeonhole Principle)

Suppose that the elements of the subset are $a_1<a_2< ...<a_s.$ Then the straightforward observations that

$\,\,$ there are $2^s-1$ non-empty subsets of the $s$ element subset

$\,\,$ the possible sums range from $a_1$ to at most $a_1+\sum_{n-s+2}^n i$

proves such a result providing $$2^s-1> \frac{(2n-s+2)(s-1)}{2}+1$$ or, equivalently, $$n<\frac{s^2-3s+2^{s+1}}{2(s-1)}.$$

This is a general result, albeit a rather weak one which can be greatly improved. I am interested in what general results can be proved for this type of problem.

EXAMPLE $s=9$.

The above result gives $n<67.375$ i.e. $n\le67$. The result of @CalvinLin (with $a=2,b=7$) improves this to $73$.

However, this bound can be greatly improved (one general method for doing this is given as an answer). Are there other methods which are even more effective for such a problem?

Answer 1

A simple way to strengthen the result is to restrict the subset of $s$ to size $[a,b]$. Common use cases are where $a = 1, 2, b = s-1, s-2$ to get rid of the extreme cases.

Then, we just require

$$ \sum_{k=a}^b { s \choose k } > bn - \frac{b^2-b}{2} - \frac{a^2+a}{2} +1 .$$

Answer 2

The following result due to Erdos is a personal favorite of mine.

Let $S$ be a finite set of positive integers such that each subset has a distinct sum. Then $$\sum_{a \in S} \frac{1}a < 2. $$

The proof can be found in this stackexchange thread.

Answer 3

Let $s=u+v+w$. (We will choose $u,v,w$ later but a choice which seems to work well is for these parameters to be roughly $\frac{s}{3},\frac{s}{6},\frac{s}{2}$.)

Consider just those subsets which have from $u$ to $s-u$ elements.

Possible sums range from $a_1+a_2+...+a_u$ to $a_{u+1}+a_{u+2}+...+a_s$ [I think here we assume that $u\le s-u$. Alex Ravsky] and so we will have two distinct subsets with the same sum unless $$a_{u+1}+a_{u+2}+...+a_s-(a_1+a_2+...+a_u)+1>\sum_{k=u}^{s-u} { s \choose k }.$$

[The left-hand side bounds the number of sums, so I think "$>$" has to be replaced by "$\ge$". Alex Ravsky]

Now consider those subsets which include at least $v$ and at most $w$ of the elements of $\{a_{u+1},a_{u+2}, ...,a_s\}$.

Possible sums range from $a_{u+1}+a_{u+2}+...+a_{u+v}$ to $a_1+a_2+...+a_u +a_{s-w+1}+a_{s-w+2}+...+a_s$ and so we will have two distinct subsets with the same sum unless $$a_1+a_2+...+a_u +a_{s-w+1}+a_{s-w+2}+...+a_s-(a_{u+1}+a_{u+2}+...+a_{u+v})+1>2^u\sum_{k=v}^w{ s-u \choose k }. $$

[I think that similarly to the above, here we assume that $v\le w$ and "$>$" has to be replaced by "$\ge$". Alex Ravsky]

Adding the two inequalities gives

$$2(a_{s-w+1}+a_{s-w+2}+...+a_s)+2>2^u\sum_{k=v}^w{ s-u \choose k }+\sum_{k=u}^{s-u} { s \choose k } $$ and therefore $$w(2n+1-w)+2>2^u\sum_{k=v}^w{ s-u \choose k }+\sum_{k=u}^{s-u} { s \choose k } $$

$$n>\frac{1}{2w}\left((w+1)(w-2)+2^u\sum_{k=v}^w{ s-u \choose k }+\sum_{k=u}^{s-u} { s \choose k } \right)$$ Therefore we will have two distinct subsets with the same sum if $$n\le \frac{1}{2w}\left((w+1)(w-2)+2^u\sum_{k=v}^w{ s-u \choose k }+\sum_{k=u}^{s-u} { s \choose k } \right)$$

EXAMPLE $s=9$.

Let $u=3,v=2,w=4$.

Then we will have two distinct subsets with the same sum unless $n>103.75$ and so there are two such subsets for $n\le103$.

Answer 4

Optimal n are the sequence https://oeis.org/A276661. The links there provide upper bounds to what the maximum such n is; you are looking for the minimum.