A simple discharging method for forbidden subposet problems

The poset $Y_{k+1,2}$ consists of $k+2$ distinct elements $x_1$, $x_2$, \dots, $x_k$, $y_1$,$y_2$, such that $x_1 \le x_2 \le \dots \le x_{k} \le y_1$,$y_2$\mathrm{.}Theposet$Y’{k+1, 2}$isthedualof$Y{k+1, 2}$Let$\rm{La}^{\sharp}(n,{Y_{k+1, 2}, Y’{k+1, 2}})$ be the size of the largest family $\mathcal{F}\subset 2^{[n]}$ that contains neither $Y_{k+1,2}$ nor $Y_{k+1,2}^{\mathrm{\prime }}$ as an induced subposet. Methuku and Tompkins proved that ${\mathrm{L}\mathrm{a}}^{\mathrm{♯}}(n,\{Y_{3,2},Y_{3,2}^{\mathrm{\prime }}\})=\mathrm{\Sigma }(n,2)$ for $n\ge 3$ and they conjectured the generalization that if $k\ge 2$ is an integer and $n \ge k+1$,then$\rm{La}^{\sharp}(n, {Y{k+1,2}, Y’_{k+1,2}}) = \Sigma(n,k)$. In this paper, we introduce a simple discharging approach and prove this conjecture.