• one can choose fifteen times five numbers from one to thirty, call each subset $M_i$

  • There exists a coloring in red and blue of the set from one to thirty of natural numbers so that each of the subsets $M_i$ would have two colors.

Mathematical Formulation

I need to define the coloring of numbers somehow. I came up with the vector notation, maybe someone could suggest something better.

  • A coloring of the set $[30]$ is a set $ F=\{x|x=(x_1,...,x_{30}), x_i\in\{0,1\}, \text{where 0=blue and 1=red} \} $

One Problem is now that I have vectors now. In F are all possible colorings of the set. If I want to get one specific $M_i$ from F, I would choose a vector and the five components out of it.

  • a k-subset is $\binom{M}{k}:=\{M\subset M||A|=k\}$

  • j=$\binom{30}{5}$ would be a subset of 5 numbers out of 30

  • $M_i=(y|y=(y_1,...y_{30}), y_l \in \{-1,0,1\}, y_l=-1 \text{ if } l \notin j, y_l=1 \text{ if } x_l=1,y_l=0 \text{ if } x_l=0 $

  • $ M_i \text{ there exists } y_k=1 \text{ and } and y_l=0 \text{ with } k,l \in j $

Any suggestion for simpler better formulation? The text is short,...

| cite | improve this question | | | | |

You should probably make sure that you somehow impose the condition that the $M_i$ are distinct (or the solution is trivial). I would then write the question as:

Denote by $\mathbb{N}^*_{\leq 30}$ the set of the first $30$ elements of $\mathbb{N}$, i.e. $\mathbb{N}^*_{\leq 30}=\{1,\dots,30\}$. For a generic set of "red numbers" $R\subseteq \mathbb{N}$, let $S_R=\{M_i\subseteq \mathbb{N}^*_{\leq 30}: (M_i\cap R), (M_i\setminus R) \neq\emptyset, |M_i|=5\}$ be the set of subsets of $\mathbb{N}^*_{\leq 30}$ with exactly $5$ elements, at least $1$ red and $1$ not. Is there an $R$ such that $|S_R|\geq 15$?

| cite | improve this answer | | | | |
  • $\begingroup$ very intresting formulation $\endgroup$ – thetha Jul 20 '18 at 13:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.