Photo Booth problem












1














There are $n$ people.
There is a Photo Booth in which they can enter at most $m$ people at one time.
They want to get a picture with all other person together.
Please solve the $F(n,m)$; minimum number of times to get required pictures.



For example, there are people: $A$, $B$, $C$ and $D$, and a Photo Booth in which they can enter at most $3$ people at one time. When they form a group $left{ A, B, C right}$, $left{B, C, D right}$ and $left{A, C, Dright}$, $A$ can get pictures with $B$, $C$ and $D$ ,etc. In this case, $F(4,3)=3$



See also Question of difficult matrix problem, minimum number of times






linear-algebra combinatorics matrices graph-theory







share|cite|improve this question



























    1














    There are $n$ people.
    There is a Photo Booth in which they can enter at most $m$ people at one time.
    They want to get a picture with all other person together.
    Please solve the $F(n,m)$; minimum number of times to get required pictures.



    For example, there are people: $A$, $B$, $C$ and $D$, and a Photo Booth in which they can enter at most $3$ people at one time. When they form a group $left{ A, B, C right}$, $left{B, C, D right}$ and $left{A, C, Dright}$, $A$ can get pictures with $B$, $C$ and $D$ ,etc. In this case, $F(4,3)=3$



    See also Question of difficult matrix problem, minimum number of times






    linear-algebra combinatorics matrices graph-theory







    share|cite|improve this question

























      1












      1








      1







      There are $n$ people.
      There is a Photo Booth in which they can enter at most $m$ people at one time.
      They want to get a picture with all other person together.
      Please solve the $F(n,m)$; minimum number of times to get required pictures.



      For example, there are people: $A$, $B$, $C$ and $D$, and a Photo Booth in which they can enter at most $3$ people at one time. When they form a group $left{ A, B, C right}$, $left{B, C, D right}$ and $left{A, C, Dright}$, $A$ can get pictures with $B$, $C$ and $D$ ,etc. In this case, $F(4,3)=3$



      See also Question of difficult matrix problem, minimum number of times






      linear-algebra combinatorics matrices graph-theory







      share|cite|improve this question













      There are $n$ people.
      There is a Photo Booth in which they can enter at most $m$ people at one time.
      They want to get a picture with all other person together.
      Please solve the $F(n,m)$; minimum number of times to get required pictures.



      For example, there are people: $A$, $B$, $C$ and $D$, and a Photo Booth in which they can enter at most $3$ people at one time. When they form a group $left{ A, B, C right}$, $left{B, C, D right}$ and $left{A, C, Dright}$, $A$ can get pictures with $B$, $C$ and $D$ ,etc. In this case, $F(4,3)=3$



      See also Question of difficult matrix problem, minimum number of times






      linear-algebra combinatorics matrices graph-theory




      linear-algebra combinatorics matrices graph-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 3 '15 at 16:36









      Soft CreamSoft Cream

      135




      135






















          1 Answer
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          active

          oldest

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          1














          A $(v,k,t)$ covering is a family of subsets of ${1,2,3dots v}$ such that each subset has $k$ elements and every subset of ${1,2,3dots,v}$ that has exactly $t$ elements is contained in one of the subsets of the family.



          What you want to find is the minimal subsets that can be in a ${v,k,2}$ covering. Here is a repository of such coverings.






          share|cite|improve this answer





















          • Thank you!!! We're struggling in this problem for ages! I didn't know such a theory.
            – Soft Cream
            Jan 3 '15 at 16:55










          • Yes, the case when $k=3$ has been much more extensively studied, The problem can be solve optimally when $v$ is congruent to $1$ or $3$ mod $6$ and it requires $frac{v(v-1)}{6}$ subsets. This is called a steiner triple system.
            – Jorge Fernández
            Jan 3 '15 at 17:12












          • Thank you! Isn't there an explicit general expression for $A(v, k, 2)$?
            – Soft Cream
            Jan 3 '15 at 17:15












          • Please see this question
            – Jorge Fernández
            Jan 4 '15 at 17:42











          Your Answer





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          1 Answer
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          active

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1














          A $(v,k,t)$ covering is a family of subsets of ${1,2,3dots v}$ such that each subset has $k$ elements and every subset of ${1,2,3dots,v}$ that has exactly $t$ elements is contained in one of the subsets of the family.



          What you want to find is the minimal subsets that can be in a ${v,k,2}$ covering. Here is a repository of such coverings.






          share|cite|improve this answer





















          • Thank you!!! We're struggling in this problem for ages! I didn't know such a theory.
            – Soft Cream
            Jan 3 '15 at 16:55










          • Yes, the case when $k=3$ has been much more extensively studied, The problem can be solve optimally when $v$ is congruent to $1$ or $3$ mod $6$ and it requires $frac{v(v-1)}{6}$ subsets. This is called a steiner triple system.
            – Jorge Fernández
            Jan 3 '15 at 17:12












          • Thank you! Isn't there an explicit general expression for $A(v, k, 2)$?
            – Soft Cream
            Jan 3 '15 at 17:15












          • Please see this question
            – Jorge Fernández
            Jan 4 '15 at 17:42
















          1














          A $(v,k,t)$ covering is a family of subsets of ${1,2,3dots v}$ such that each subset has $k$ elements and every subset of ${1,2,3dots,v}$ that has exactly $t$ elements is contained in one of the subsets of the family.



          What you want to find is the minimal subsets that can be in a ${v,k,2}$ covering. Here is a repository of such coverings.






          share|cite|improve this answer





















          • Thank you!!! We're struggling in this problem for ages! I didn't know such a theory.
            – Soft Cream
            Jan 3 '15 at 16:55










          • Yes, the case when $k=3$ has been much more extensively studied, The problem can be solve optimally when $v$ is congruent to $1$ or $3$ mod $6$ and it requires $frac{v(v-1)}{6}$ subsets. This is called a steiner triple system.
            – Jorge Fernández
            Jan 3 '15 at 17:12












          • Thank you! Isn't there an explicit general expression for $A(v, k, 2)$?
            – Soft Cream
            Jan 3 '15 at 17:15












          • Please see this question
            – Jorge Fernández
            Jan 4 '15 at 17:42














          1












          1








          1






          A $(v,k,t)$ covering is a family of subsets of ${1,2,3dots v}$ such that each subset has $k$ elements and every subset of ${1,2,3dots,v}$ that has exactly $t$ elements is contained in one of the subsets of the family.



          What you want to find is the minimal subsets that can be in a ${v,k,2}$ covering. Here is a repository of such coverings.






          share|cite|improve this answer












          A $(v,k,t)$ covering is a family of subsets of ${1,2,3dots v}$ such that each subset has $k$ elements and every subset of ${1,2,3dots,v}$ that has exactly $t$ elements is contained in one of the subsets of the family.



          What you want to find is the minimal subsets that can be in a ${v,k,2}$ covering. Here is a repository of such coverings.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 3 '15 at 16:44









          Jorge FernándezJorge Fernández

          75.1k1190191




          75.1k1190191












          • Thank you!!! We're struggling in this problem for ages! I didn't know such a theory.
            – Soft Cream
            Jan 3 '15 at 16:55










          • Yes, the case when $k=3$ has been much more extensively studied, The problem can be solve optimally when $v$ is congruent to $1$ or $3$ mod $6$ and it requires $frac{v(v-1)}{6}$ subsets. This is called a steiner triple system.
            – Jorge Fernández
            Jan 3 '15 at 17:12












          • Thank you! Isn't there an explicit general expression for $A(v, k, 2)$?
            – Soft Cream
            Jan 3 '15 at 17:15












          • Please see this question
            – Jorge Fernández
            Jan 4 '15 at 17:42


















          • Thank you!!! We're struggling in this problem for ages! I didn't know such a theory.
            – Soft Cream
            Jan 3 '15 at 16:55










          • Yes, the case when $k=3$ has been much more extensively studied, The problem can be solve optimally when $v$ is congruent to $1$ or $3$ mod $6$ and it requires $frac{v(v-1)}{6}$ subsets. This is called a steiner triple system.
            – Jorge Fernández
            Jan 3 '15 at 17:12












          • Thank you! Isn't there an explicit general expression for $A(v, k, 2)$?
            – Soft Cream
            Jan 3 '15 at 17:15












          • Please see this question
            – Jorge Fernández
            Jan 4 '15 at 17:42
















          Thank you!!! We're struggling in this problem for ages! I didn't know such a theory.
          – Soft Cream
          Jan 3 '15 at 16:55




          Thank you!!! We're struggling in this problem for ages! I didn't know such a theory.
          – Soft Cream
          Jan 3 '15 at 16:55












          Yes, the case when $k=3$ has been much more extensively studied, The problem can be solve optimally when $v$ is congruent to $1$ or $3$ mod $6$ and it requires $frac{v(v-1)}{6}$ subsets. This is called a steiner triple system.
          – Jorge Fernández
          Jan 3 '15 at 17:12






          Yes, the case when $k=3$ has been much more extensively studied, The problem can be solve optimally when $v$ is congruent to $1$ or $3$ mod $6$ and it requires $frac{v(v-1)}{6}$ subsets. This is called a steiner triple system.
          – Jorge Fernández
          Jan 3 '15 at 17:12














          Thank you! Isn't there an explicit general expression for $A(v, k, 2)$?
          – Soft Cream
          Jan 3 '15 at 17:15






          Thank you! Isn't there an explicit general expression for $A(v, k, 2)$?
          – Soft Cream
          Jan 3 '15 at 17:15














          Please see this question
          – Jorge Fernández
          Jan 4 '15 at 17:42




          Please see this question
          – Jorge Fernández
          Jan 4 '15 at 17:42


















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