Non-star connected graphs H with m edges satisfy hat R_r(H) = Omega(r^2 m), with stars the sole exception; a similar quadratic lower bound holds for non-star trees using the beta parameter.
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A note on the multicolor size-Ramsey numbers of connected graphs
Non-star connected graphs H with m edges satisfy hat R_r(H) = Omega(r^2 m), with stars the sole exception; a similar quadratic lower bound holds for non-star trees using the beta parameter.