Let $F $ be a subset family of the set {$ 1, 2, ..., 2017 $} such that for any $ A, B \in F $, worth that $A \cap B$ has exactly one element. Determine as many as possible of $ F $ elements

Solution: Generalization: if the total number set is $\{1, 2, ..., n\}$, then the maximum of $|F|$ is $n$. In the original problem $n=2017$, so $max|F|=\boxed{2017}$.

  1. We claim that $|F| \leq n$ Consider a map from one subset $A$ of $\{1, 2, ..., n\}$ to a $n$-dimension vector $V=(v_1, v_2, ..., v_n)^T$. For $1 \leq i \leq n$, if $i \in A$ then $v_i=1$, else $v_i=0$. Consider the vector set mapped from $F$: $\{V_1, V_2, ..., V_m\}$. We could prove the vectors are linearly independent. For $i \ne j$, $<V_i, V_j>=V_i^TV_j=1$, since there is exactly one element included in any two elements of $F$. For $i = j$, $<V_i, V_i>=|V_i| \geq 1$, where $|V_i|$ counts the number of $1$ appeared in $V_i$. Consider $S=\sum_{k=1}^{m} a_kV_k$, if $S=(0,0,...,0)^T$, then $<S, S>=0$. However, $<S, S>=\sum_{i=1}^{m} a_i^2<V_i, V_i>+\sum_{i \ne j} 2a_ia_j<V_i, V_j>=(\sum_{i=1}^{m} a_i)^2+\sum_{i=1}^{m} a_i^2(|V_i|-1) \geq 0$. So the equality holds only when $a_i=0$, that means $\{V_1, V_2, ..., V_m\}$ are linearly independent, so that $m \leq n$, so that $|F| \leq n$.

  2. A construction of $|F|=n$ Consider $F=\{\{a,n\} | 1 \leq a \leq n-1\} \cup\{n\}$. $|F|=n$, and any $ A, B \in F$, $A \cap B=\{n\}$.

I didn't understand the logic of this solution

  • $\begingroup$ what does "worth that" mean? $\endgroup$ – coffeemath Nov 19 '19 at 23:52
  • $\begingroup$ What part don't you get? Just go through the argument line by line. $\endgroup$ – lulu Nov 19 '19 at 23:57

We can rewrite the argument as follows.

(1). Each set in $F$ can be represented as a vector. For example, $\{2,3,6\}$ is represented by $$(0,1,1,0,0,1,0,0,...)$$ where a "$1$" in the $i$th position indicates that the set contains "$i$".

If $|F|=m$, then we now have $m$ such vectors $$V_1,V_2,V_3,...V_m.$$ The crucial fact is that for any pair of these vectors, there is one and only one position where they both have a "$1$".

Now suppose we could find numbers $a_1,a_2,a_3,...a_m$ such that $$W=\sum_1^m a_iV_i=0.$$

Then the scalar product of $W$ with itself is, of course, zero.

We also know that if $i\ne j$, then $V_i.V_j=1,$ whereas $V_i.V_i$ is the number of "$1$"s in $V_i$ which we can denote by $||V_i||$. Note that each $||V_i||\ge 1$ and only one $||V_i||$ can equal $1$.

The scalar product of $W$ with itself can be considered to be the sum of lots of scalar products of the form $a_iV_i.a_jV_j$. Summing these we obtain $$0=\sum_1^m a_i^2||V_i||+\sum_{i\ne j}a_ia_j=\sum_1^m a_i^2(||V_i||-1)+\sum_1^m a_i^2+\sum_{i\ne j}a_ia_j$$

$$=\sum_1^m a_i^2(||V_i||-1)+(a_1+a_2+... +a_m)^2.$$

The only possibility is that each $a_i=0$. The vectors $V_i$ are therefore linearly independent and therefore there can be at most $n$ of them i.e. $|F|\le n$.

(2). The upper bound of $n$ can be attained since these $n$ sets satisfy the conditions:- $$\{1,n\},\{2,n\},\{3,n\},...\{n-1,n\},\{n\}.$$

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How is your linear algebra? It is saying we are constructing vectors in a 2017-dimensional space. I will give the example in $3$-D. Let $\{1,2,3\}$ be the base set, so the subset $\{1,3\}$ maps to the vector $\begin{bmatrix} 1 \\ 0 \\ 1 \\ \end{bmatrix}$. Look at what happens if we have linear dependence (this means a "good" combination of the non-zero vectors becomes zero) for example:

$\begin{bmatrix} 1 \\ 1 \\ 0 \\ \end{bmatrix}-\begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix}-\begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix}=0$. Now see that this is like saying in terms of our original sets $\{1,2\} -\{1\} - \{2\} = \emptyset$. But notice that $|\{1,2\}\cap \{1\}|=1, |\{1,2\}\cap \{2\}|=1,$ BUT $|\{1\}\cap \{2\}|=0$.

Try to extend this idea of linear dependence requiring the sets to over or under intersect to the full argument given.

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