Translating and inflating a set of $k$-dimensional subspaces of $mathbb F_p^n$ to form a cover by affine hyperplanes?

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Fix a prime number $p$ and consider the affine space $V = mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, ldots, V_n subseteq V$ of dimension $k$, and take $v_i notin V_i$. Do there always exist subspaces $W_i supseteq V_i$ of codimension $1$ and $t_1, ldots, t_n in V$ such that $v_i notin W_i$ and
$$t_1 + W_1, ldots, t_n + W_n$$
is a cover of $V$ by affine hyperplanes? Is this true for at least some $n$?



Note that:



  • Comparing cardinalities, one needs $n geq p$.

  • It suffices to do the case where $k = lfloor fracn-12 rfloor$ is the largest integer strictly less than $n/2$ (we can always replace the $V_i$ by larger subspaces)

I'm especially interested if there exists an even $n > p$ for which this is true. Also, I don't mind assuming that $p$ belongs to some infinite set.



Motivation: If it is true for some even $n$, this would imply that the answer to this question is 'yes':
Hat 'trick': Can one of them guess right?




linear-algebra combinatorics finite-fields affine-geometry finite-geometry




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    Fix a prime number $p$ and consider the affine space $V = mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, ldots, V_n subseteq V$ of dimension $k$, and take $v_i notin V_i$. Do there always exist subspaces $W_i supseteq V_i$ of codimension $1$ and $t_1, ldots, t_n in V$ such that $v_i notin W_i$ and
    $$t_1 + W_1, ldots, t_n + W_n$$
    is a cover of $V$ by affine hyperplanes? Is this true for at least some $n$?



    Note that:



    • Comparing cardinalities, one needs $n geq p$.

    • It suffices to do the case where $k = lfloor fracn-12 rfloor$ is the largest integer strictly less than $n/2$ (we can always replace the $V_i$ by larger subspaces)

    I'm especially interested if there exists an even $n > p$ for which this is true. Also, I don't mind assuming that $p$ belongs to some infinite set.



    Motivation: If it is true for some even $n$, this would imply that the answer to this question is 'yes':
    Hat 'trick': Can one of them guess right?




    linear-algebra combinatorics finite-fields affine-geometry finite-geometry




    share|cite|improve this question























      up vote
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      down vote

      favorite









      up vote
      4
      down vote

      favorite











      Fix a prime number $p$ and consider the affine space $V = mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, ldots, V_n subseteq V$ of dimension $k$, and take $v_i notin V_i$. Do there always exist subspaces $W_i supseteq V_i$ of codimension $1$ and $t_1, ldots, t_n in V$ such that $v_i notin W_i$ and
      $$t_1 + W_1, ldots, t_n + W_n$$
      is a cover of $V$ by affine hyperplanes? Is this true for at least some $n$?



      Note that:



      • Comparing cardinalities, one needs $n geq p$.

      • It suffices to do the case where $k = lfloor fracn-12 rfloor$ is the largest integer strictly less than $n/2$ (we can always replace the $V_i$ by larger subspaces)

      I'm especially interested if there exists an even $n > p$ for which this is true. Also, I don't mind assuming that $p$ belongs to some infinite set.



      Motivation: If it is true for some even $n$, this would imply that the answer to this question is 'yes':
      Hat 'trick': Can one of them guess right?




      linear-algebra combinatorics finite-fields affine-geometry finite-geometry




      share|cite|improve this question













      Fix a prime number $p$ and consider the affine space $V = mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, ldots, V_n subseteq V$ of dimension $k$, and take $v_i notin V_i$. Do there always exist subspaces $W_i supseteq V_i$ of codimension $1$ and $t_1, ldots, t_n in V$ such that $v_i notin W_i$ and
      $$t_1 + W_1, ldots, t_n + W_n$$
      is a cover of $V$ by affine hyperplanes? Is this true for at least some $n$?



      Note that:



      • Comparing cardinalities, one needs $n geq p$.

      • It suffices to do the case where $k = lfloor fracn-12 rfloor$ is the largest integer strictly less than $n/2$ (we can always replace the $V_i$ by larger subspaces)

      I'm especially interested if there exists an even $n > p$ for which this is true. Also, I don't mind assuming that $p$ belongs to some infinite set.



      Motivation: If it is true for some even $n$, this would imply that the answer to this question is 'yes':
      Hat 'trick': Can one of them guess right?





      linear-algebra combinatorics finite-fields affine-geometry finite-geometry





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      share|cite|improve this question




      share|cite|improve this question








      edited Aug 7 at 19:00
























      asked Aug 7 at 18:35









      barto

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