Komlós properties in Banach lattices

Several Komlós like properties in Banach lattices are investigated. We prove that C(K) fails the oo\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${oo}$$\end{document}-pre-Komlós property, assuming that the compact Hausdorff space K has a nonempty separable open subset U without isolated points such that every u∈\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in}$$\end{document}U has countable neighborhood base. We prove also that, for any infinite dimension al Banach lattice E, there is an unbounded convex uo-pre-Komlós set C⊆E+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\subseteq E_{+}}$$\end{document} which is not uo-Komlós.