How many n-colour points are needed to force a regularly-spaced set of one colour?

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As part of a proof in finding the minimum coloured grid that is guaranteed to have some four points that form an aligned square of one colour, I formed a technique that requires finding the smallest line of $n$-coloured points, or equivalently the shortest string of $n$ different letters, that is guaranteed to have a regularly-spaced set of the same colour (letter).



For two colours and looking for a regularly spaced set of three points, we require $9$ points as shown in the following cases, forced from the initial patterns (shown red):



$mathtt colorredOOXOOXX?$

$mathtt colorredOOXXOOX?X$

$mathtt colorredOXOOXOXX?$

$mathtt colorredOXOXXOXO?$

$mathtt colorredOXXOOXXO?$



Where in each case the $mathtt?$ creates a regularly-spaced triplet whether filled with $mathttO$ or $mathttX$. For example in the final case we produce either a regular set of $mathttX$s on a 3-step or a regular set of $mathttO$s on a 4-step pattern.



Is there a method of finding a bound on the number of total points required to be sure of a regularly-spaced set of $k$ points all of the same colour in an $n$-colouring?




combinatorics number-theory ramsey-theory




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  • 4




    Van der Waerden's theorem
    – bof
    Jul 23 at 6:57










  • @bof - ouch. a very tricky problem then. Limited answers very welcome.
    – Joffan
    Jul 23 at 7:02














up vote
5
down vote

favorite
3












As part of a proof in finding the minimum coloured grid that is guaranteed to have some four points that form an aligned square of one colour, I formed a technique that requires finding the smallest line of $n$-coloured points, or equivalently the shortest string of $n$ different letters, that is guaranteed to have a regularly-spaced set of the same colour (letter).



For two colours and looking for a regularly spaced set of three points, we require $9$ points as shown in the following cases, forced from the initial patterns (shown red):



$mathtt colorredOOXOOXX?$

$mathtt colorredOOXXOOX?X$

$mathtt colorredOXOOXOXX?$

$mathtt colorredOXOXXOXO?$

$mathtt colorredOXXOOXXO?$



Where in each case the $mathtt?$ creates a regularly-spaced triplet whether filled with $mathttO$ or $mathttX$. For example in the final case we produce either a regular set of $mathttX$s on a 3-step or a regular set of $mathttO$s on a 4-step pattern.



Is there a method of finding a bound on the number of total points required to be sure of a regularly-spaced set of $k$ points all of the same colour in an $n$-colouring?




combinatorics number-theory ramsey-theory




share|cite|improve this question















  • 4




    Van der Waerden's theorem
    – bof
    Jul 23 at 6:57










  • @bof - ouch. a very tricky problem then. Limited answers very welcome.
    – Joffan
    Jul 23 at 7:02












up vote
5
down vote

favorite
3









up vote
5
down vote

favorite
3






3





As part of a proof in finding the minimum coloured grid that is guaranteed to have some four points that form an aligned square of one colour, I formed a technique that requires finding the smallest line of $n$-coloured points, or equivalently the shortest string of $n$ different letters, that is guaranteed to have a regularly-spaced set of the same colour (letter).



For two colours and looking for a regularly spaced set of three points, we require $9$ points as shown in the following cases, forced from the initial patterns (shown red):



$mathtt colorredOOXOOXX?$

$mathtt colorredOOXXOOX?X$

$mathtt colorredOXOOXOXX?$

$mathtt colorredOXOXXOXO?$

$mathtt colorredOXXOOXXO?$



Where in each case the $mathtt?$ creates a regularly-spaced triplet whether filled with $mathttO$ or $mathttX$. For example in the final case we produce either a regular set of $mathttX$s on a 3-step or a regular set of $mathttO$s on a 4-step pattern.



Is there a method of finding a bound on the number of total points required to be sure of a regularly-spaced set of $k$ points all of the same colour in an $n$-colouring?




combinatorics number-theory ramsey-theory




share|cite|improve this question











As part of a proof in finding the minimum coloured grid that is guaranteed to have some four points that form an aligned square of one colour, I formed a technique that requires finding the smallest line of $n$-coloured points, or equivalently the shortest string of $n$ different letters, that is guaranteed to have a regularly-spaced set of the same colour (letter).



For two colours and looking for a regularly spaced set of three points, we require $9$ points as shown in the following cases, forced from the initial patterns (shown red):



$mathtt colorredOOXOOXX?$

$mathtt colorredOOXXOOX?X$

$mathtt colorredOXOOXOXX?$

$mathtt colorredOXOXXOXO?$

$mathtt colorredOXXOOXXO?$



Where in each case the $mathtt?$ creates a regularly-spaced triplet whether filled with $mathttO$ or $mathttX$. For example in the final case we produce either a regular set of $mathttX$s on a 3-step or a regular set of $mathttO$s on a 4-step pattern.



Is there a method of finding a bound on the number of total points required to be sure of a regularly-spaced set of $k$ points all of the same colour in an $n$-colouring?





combinatorics number-theory ramsey-theory





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 23 at 6:12









Joffan

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  • 4




    Van der Waerden's theorem
    – bof
    Jul 23 at 6:57










  • @bof - ouch. a very tricky problem then. Limited answers very welcome.
    – Joffan
    Jul 23 at 7:02












  • 4




    Van der Waerden's theorem
    – bof
    Jul 23 at 6:57










  • @bof - ouch. a very tricky problem then. Limited answers very welcome.
    – Joffan
    Jul 23 at 7:02







4




4




Van der Waerden's theorem
– bof
Jul 23 at 6:57




Van der Waerden's theorem
– bof
Jul 23 at 6:57












@bof - ouch. a very tricky problem then. Limited answers very welcome.
– Joffan
Jul 23 at 7:02




@bof - ouch. a very tricky problem then. Limited answers very welcome.
– Joffan
Jul 23 at 7:02















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