Denote strings $u,v$ from $\{0,1\}^n$.
Denote concatenated pair $[uv]$.
Denote $$[uv]_{1}=\{[uv]\oplus e_i\}_{i=1}^{2n}$$ collection of pairs with Hamming distance $1$ from $[uv]$ string concatenation.
$e_i\in\{0,1\}^{2n}$ is almost $0$ vector with $1$ at $i$th position.
Denote $\mathsf{G_{2n}}$ to be universe of deterministic games played by Alice, Bob where Alice gets a random string $u$ from $\{0,1\}^n$, Bob gets a random string $v$ from $\{0,1\}^n$ where games in universe either ends in win by one side or draw (games containing drawing a possibility).
Take values $c\geq1$, $r>2$ fixed.
Is is true that $\forall n\in\Bbb N$, there is no game $G\in\mathsf{G_{2n}}$ such that following conditions will be satisfied?
$1.$ If at every string pair $u,v$ that results in a win, number of pairs from collection $[uv]_1$ that results in a draw is atmost $n^{\frac{1}{rc}}$.
$2.$ If at every string pair $u,v$ that results in a draw, number of pairs from collection $[uv]_1$ that results in a win is atmost $n^{\frac{1}{rc}}$.
$3.$ There is a pair $u,v$ that results in a win (or draw) such that balls of vectors with hamming distance atleast $n^{\frac{1}{c}}$ from $u,v$ respectively contains atleast a pair of $u,v$ that results in a draw (or win).
