Given $n$ vectors $v_1,\dots,v_n\in\Bbb R^N$ with $\|v_i\|_2^2\leq1$ at every $i\in\{1,\dots,n\}$, Komlos conjecture states that, there is a $c\in\Bbb R$ (independent of $n,N$) such that at some $\epsilon\in\{-1,+1\}^n$, $$\Big\|\sum_{i=1}^n\epsilon_iv_i\Big\|_\infty<c.$$ What is best lower bound known for $c$?Best upper bound is $c=O(\sqrt{\log n})$.Is there a set of examples for $c=1+\delta$ for any $\delta>0$ where Komlos conjecture fails?
Constant in Komlos conjecture
co.combinatorics;extremal combinatorics