# 4-tuples with close sums

Some $$4$$-tuples of positive real numbers $$(a_1,b_1,c_1,d_1),\dots,(a_n,b_n,c_n,d_n)$$ are given, with all $$a_i,b_i,c_i,d_i\leq 1$$. Is it always possible to partition $$\{1,2,\dots,n\}$$ into two subsets $$X,Y$$ so that at least $$3$$ of the following $$4$$ inequalities hold:

$$\left|\sum_Xa_i-\sum_Ya_i\right|\leq 1, \left|\sum_Xb_i-\sum_Yb_i\right|\leq 1$$ $$\left|\sum_Xc_i-\sum_Yc_i\right|\leq 1, \left|\sum_Xd_i-\sum_Yd_i\right|\leq 1$$

An affirmative answer here would imply one for this (unsolved) question.

• As you are asking for "3 out of 4", have you found an example where there is no partition such that all 4 inequalities hold? – Wolfgang May 19 '19 at 20:06
• Yes, for example $(1,1,0,0),(1,0,1,0),(0,1,1,0)$ – pi66 May 19 '19 at 21:13