A Stability Refinement of Shearer's Bound for Triangle-Free Graphs

We begin by briefly exploring a restricted graph family of triangle free graphs to establish a baseline to work the rest of the paper off. We then develop a quantitative lower bound on the discrete curvature of Shearer's bound, yielding a stability refinement. In the Ramsey-scale regime, we derive a deficit-sensitive bound on the degree variance and show that near-saturation of Shearer’s bound forces degree concentration. We will then determine that two positive-density vertex populations whose degrees differ by a fixed positive proportion of k necessarily produce a linear Shearer deficit, ruling out macroscopic degree splitting in the near-saturated regime. We also obtain a lower bound on the average degree from Ramsey-scale order and extend the resulting structural control to a four-cycle abundance theorem in the tight degree-size regime. These results provide a quantitative stability framework for studying triangle-free graphs near the R(3,k) scale.