Classification conjecture for maximum (d,t)(d,t)-EKR sets in dual polar graphs

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Let YY be a maximum-size (d,t)(d,t)-EKR set of generators in a polar space of rank dd and type parameter ee, meaning that any two members of YY meet in dimension at least d−td-t. A maximum (d,t)(d,t)-EKR set classification conjecture. One of the following cases occurs:

  1. YY is the set of all generators on a fixed (d−t)(d-t)-space.
  2. tt is even, and YY is the set of all generators meeting a fixed generator in dimension at least d−t2d-\frac{t}{2}.
  3. tt is odd, and YY is the set of all generators meeting a fixed (d−1)(d-1)-dimensional space in dimension at least d−t2−12d-\frac{t}{2}-\frac{1}{2}.
  4. e=0e=0, t=d−1t=d-1, dd is odd, and YY is the largest example for Q+(2d−1,q)Q^+(2d-1,q) described by the cited reference.

This conjecture seeks a complete classification of maximum (d,t)(d,t)-EKR sets, extending known classifications in special cases. The proposed families arise from the natural geometric configurations of generators; the general classification remains open in the source.

References

Primary source

Ferdinand Ihringer and Klaus Metsch, “Large \0, 1, , t\-Cliques in Dual Polar Graphs”, arXiv:1510.01697 (2015).

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