Given $n\in\Bbb N$, $\alpha>0$, $\beta\in\big[\frac12,1\big]$ denote $\mathcal R_{n,\alpha,\beta}$ as collection of all $2^n\times 2^{n^\alpha}$ $0/1$ matrices with every row summing to **strictly greater** than $\beta2^{n^\alpha}$.

Denote $f(\mathcal R_{n,\alpha,\beta})$ as worst case minimum number of columns among matrices in $\mathcal R_{n,\alpha,\beta}$ that is needed so that in any such matrix there is always a collection $\tau\leq f(\mathcal R_{n,\alpha,\beta})$ of columns which when considered as a submatrix has every row with **strictly greater** than $\frac\tau2$ entries $1$.

(Clearly such a submatrix exists for every matrix in $\mathcal R_{n,\alpha,\beta}$ as every $2^n\times 2^{n^\alpha}$ matrix in $\mathcal R_{n,\alpha,\beta}$ has this property).

$1.$ Does ${f\Big(\mathcal R_{n,\alpha,\frac34}\Big)}=O(n^{\alpha\gamma})$ hold at *some fixed* $\gamma\geq0$?

$2.$ Does ${f\Big(\mathcal R_{n,\alpha,\frac12+\frac1{g(n)}}\Big)}=O(n^{\alpha\gamma''})$ hold at *some fixed* $\gamma''\geq0$ for every $g:\Bbb R\rightarrow\Bbb R$ with $x<y\implies g(x)\leq g(y)$ with $g(n)\leq2^{n^\zeta}$ where $\zeta\in[1,\alpha]$ holds or does any such $\gamma''$ depend on $g$ **OR** for any such growing $g$ does
${f\Big(\mathcal R_{n,\alpha,\frac12+\frac1{g(n)}}\Big)}=\omega(n^{\alpha\delta'})$
hold at *any fixed* $\delta'>0$ hold?

**Clearly if $g(n)>2^{n^\alpha}$ the scenario reduces to case of $\frac12$ more or less morally since number of columns scales much more fast than probability of getting a balance of more $1$s.**

Are there any references and any good tools to study this problem?

Without $\tau$ criterion even if $\beta=\frac12$ just ${n^\alpha} + 1$ columns suffice (we can use greedy method by picking a column which has most number of $1$s and removing this column and collection of rows which are $1$ at this column and reapply greedy method. It is clear that if $\alpha\geq1$ there should be a column with at least $\frac12$ number of $1$s).