Necessary conditions for binomial collisions

Abstract

We shall give some necessary conditions for the equation $\binom{x}{a}=\binom{y}{b}$ to hold: if $\binom{2n+\delta}{n-m}=\binom{2n+l}{n-k}$ with $\delta=0$ or $1$, $0<m\leq 0.735k$, $k<n$ and $n$ sufficiently large, then $l>(cn/\log n)^{40/21}$ for some constant $c$.