# From text to mathematical formulation, combinatorics, coloring

Formulation:

• 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,...

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 $|S_R|\geq 15$?