Let \(H^+\) be closed half-space in \(\mathbb{R}^n\), \(H\) be a corresponding hyperplane, and \(S\) be a unit sphere in \(H^+\) touching \(H\). The one-sided kissing number \(B(n)\) is the maximal number of unit nonoverlapping spheres in \(H^+\) that can touch \(S\). It is clear that \(B(2)=4\), it was proved by G. Fejes Tóth in [Period. Math. Hung. 12, 125–127 (1981; Zbl 0438.52014)] that \(B(3)=9\), and by K. Bezdek in [Eur. J. Comb. 27, No. 6, 864-883 (2006; Zbl 1091.52010)] that \(B(4)\) is either \(18\) or \(19\). K. Bezdek has also conjectured that \(B(4)=18\). In the present paper, the author proves Bezdek’s conjecture. The proof is based on the author’s extension of Delsarte’s method (see e.g. P. Delsarte, J. M. Goethals, J. J. Seidel, [Geom. Dedicata 6, 363–388 (1977; Zbl 0376.05015)]). In the last section the author discusses relations between \(B(n)\) and the kissing number \(k(n)\) (the highest number of nonoverlapping unit spheres in \(\mathbb{R}^n\) that can touch a given unit sphere). Noticing that \(B(n)=K(n):= (k(n-1)+k(n))/2\) for \(n=2, 3, 4\), the author conjectures that \(B(n)=K(n)\) for \(n=5, 8, 24\) and that the ratio \(B(n)/K(n)\) tends to one as \(n\) grows to infinity.