Number of $k$-sets in $[n]$ so that any two of them share at most 2 elements?












2












$begingroup$


Let $[n]$ be ${1,2,dots,n}$. Let
begin{align}T_ell:=Big|Big{&{K_1,dots,K_m} text{ is "maximal"}:\ &text{each $K_i$ is a $k$-subset of $[n]$, and $|K_icap K_j|le ell$ for all $1le i<jle m$} Big}Big|,end{align}
where "maximal" means you cannot find one more $k$-subset $K_{m+1}$ so that ${K_1,dots,K_{m+1}} text{ satisfies that each $K_i$ is a $k$-subset of $[n]$, and $|K_icap K_j|le ell$ for all $1le i<jle m+1$}$.


We can assume $1ll kll n$. I am interested in an upper bound on $T_2$. It seems this problem is related to partition, especially when $ell=0$ ?






combinatorics number-theory discrete-mathematics







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$endgroup$

















    2












    $begingroup$


    Let $[n]$ be ${1,2,dots,n}$. Let
    begin{align}T_ell:=Big|Big{&{K_1,dots,K_m} text{ is "maximal"}:\ &text{each $K_i$ is a $k$-subset of $[n]$, and $|K_icap K_j|le ell$ for all $1le i<jle m$} Big}Big|,end{align}
    where "maximal" means you cannot find one more $k$-subset $K_{m+1}$ so that ${K_1,dots,K_{m+1}} text{ satisfies that each $K_i$ is a $k$-subset of $[n]$, and $|K_icap K_j|le ell$ for all $1le i<jle m+1$}$.


    We can assume $1ll kll n$. I am interested in an upper bound on $T_2$. It seems this problem is related to partition, especially when $ell=0$ ?






    combinatorics number-theory discrete-mathematics







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      Let $[n]$ be ${1,2,dots,n}$. Let
      begin{align}T_ell:=Big|Big{&{K_1,dots,K_m} text{ is "maximal"}:\ &text{each $K_i$ is a $k$-subset of $[n]$, and $|K_icap K_j|le ell$ for all $1le i<jle m$} Big}Big|,end{align}
      where "maximal" means you cannot find one more $k$-subset $K_{m+1}$ so that ${K_1,dots,K_{m+1}} text{ satisfies that each $K_i$ is a $k$-subset of $[n]$, and $|K_icap K_j|le ell$ for all $1le i<jle m+1$}$.


      We can assume $1ll kll n$. I am interested in an upper bound on $T_2$. It seems this problem is related to partition, especially when $ell=0$ ?






      combinatorics number-theory discrete-mathematics







      share|cite|improve this question









      $endgroup$




      Let $[n]$ be ${1,2,dots,n}$. Let
      begin{align}T_ell:=Big|Big{&{K_1,dots,K_m} text{ is "maximal"}:\ &text{each $K_i$ is a $k$-subset of $[n]$, and $|K_icap K_j|le ell$ for all $1le i<jle m$} Big}Big|,end{align}
      where "maximal" means you cannot find one more $k$-subset $K_{m+1}$ so that ${K_1,dots,K_{m+1}} text{ satisfies that each $K_i$ is a $k$-subset of $[n]$, and $|K_icap K_j|le ell$ for all $1le i<jle m+1$}$.


      We can assume $1ll kll n$. I am interested in an upper bound on $T_2$. It seems this problem is related to partition, especially when $ell=0$ ?






      combinatorics number-theory discrete-mathematics




      combinatorics number-theory discrete-mathematics






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Feb 1 at 17:30









      ConnorConnor

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