Nondivisible complement-of-Kneser policies retain a capacity gap

Exact statement

For the complete $r$ -uniform policy, $J_\psi=\overline{KG_{n,r}}$ . If $r\mid n$ , its capacity is $n/r$ ; if $r\nmid n$ , the one-shot packing $\lfloor n/r\rfloor$ and fractional/Lovász ceiling $n/r$ diverge. The first complete two-support case is $T_5=\overline{KG_{5,2}}$ , with published values $\alpha(T_5^{\boxtimes d})=2,5,12,27$ for $d=1,2,3,4$ .

StatusExact reduction to the published triangular-graph capacity problem; no new capacity bound claimed

External reviewThe triangular-graph bounds are published; the FPRD policy translation has no external review.

Context

Kneser graphs themselves have capacity equal to their EKR independence number; their nondivisible complements retain a genuine one-shot-to-asymptotic gap.

Proof or evidence

The complete r-support policy has the complement Kneser overlap graph. For T5, supermultiplicativity gives 12*5=60 at power five and the standard recursive theta bound gives floor((5/2)*27)=67.

Complement-of-Kneser boundary →

Verification notes

The complete two-support policy on five participants is verified to have the 10-vertex, 6-regular triangular conflict graph T5.

Limitations

The exact value of alpha(T5 strong-power 5) is not determined here.
The finite-attainment question is not open: nonattaining graphs have been known since 1990 and recent connected families are published.

Open work

Determine or tighten 60 <= alpha(T5 strong-power 5) <= 67 with symmetry-reduced XCC/DLX.