Vanishing ratio of mixed and classical Ramsey numbers

Prove that, for each fixed integer m greater than or equal to 4, the ratio t(m,n)/r(m,n) tends to zero as n tends to infinity, where t(m,n) is the mixed Ramsey number and r(m,n) is the classical Ramsey number.

Background

The mixed Ramsey number t(m,n) is the least N such that every red-blue coloring of the edges of K_N contains either an m-element irredundant set in the blue graph or an n-element independent set in the red graph. The classical Ramsey number r(m,n) is defined similarly, with an m-element blue independent set in place of an irredundant set.

The paper notes that Chen, Hattingh, and Rousseau established t(3,n)/r(3,n) tending to zero, while their bounds did not settle the analogous statement for every fixed m greater than or equal to 4. The paper verifies the conjecture only for m=4 by proving that t(4,n)/r(4,n) tends to zero.