The two big problems in combinatorics and theoretical computer science
In this post, we show Lemma 2.1 of Step 2 that proves (4.5): \( |\sigma_-(\mathcal{M})| < 3^\beta \gamma |\sigma_+(\mathcal{N}) ||\) seeing the description figure given last time. Links to: 6/22/26 for (3.x), 6/25/26 for (4.x). Links for mobile devices
We are in Step 2 of our proof to construct some families of sequenced neighbor pairs. Links to: 6/18/26 for (2.x), 6/22/26 for (3.x). Figure: the three relations (4.7)-(4.9) to show (4.5) In my past tries, no straightforward criterion such as (2.7) can contain the big subtlety completely without a loophole. The extra twist here […]
In this post, we check the first step of our proof given last time. Links to: 6/15/26 for (1.x), 6/18/26 for (2.x). Links for mobile devices
Based on the main idea given last time, we change the object domain from \( \mathcal{F} \) to \( \mathcal{F}^2 \) to find a good sunflower core \( C \) of a desired \(k\)-sunflower in \( \mathcal{F} \). We continue to overview this approach. Links for mobile devices
Last year, I was working on Paper 1-2 uploaded on arXiv showing on the left. It tries to prove that \( \mathcal{F} \subset {X \choose m} \) includes a \( k\)-sunflower if \( |\mathcal{F}| \ge \left( \frac{c k^2 \ln m}{\ln \ln m} \right)^m \). The current best-known result is \( (c \ln m)^m \) for […]
We now show Theorem 2.3: \( \kappa[Shd(\mathcal{F}, r)] < \frac{4 r}{m} \kappa(\mathcal{F}) + \frac{m^2}{n} \) for any \( m \in [\epsilon n] \) and \( r \in [m] \). Link to: 6/22/23 Like we saw on 7/21/25, the theorem means that the \( r \)-shadow is a vast majority of \( {X \choose r} \) […]
With the double inequality (1.1) on 7/21/25 now available, let’s prove Theorem 2.1: \( \kappa[ Shd (\mathcal{F}, r) ] < \frac{r}{m} \kappa(\mathcal{F}) + \frac{m^2}{n} \) for powers \( m, r \) of 2 in this post. We first confirm it when \( r=m/2 \). It’s a case to pivot all other \( r \) as […]
Let’s finish proving Corollary A.5 in this post to confirm the double inequality (1.1) on 7/21/25. Links to: 8/11/25, 6/16/25. Links for mobile devices
To continue our proof of Lemma A.4 and Corollary A.5, we see the integral given in (3.3) last time equals \[ \int_0^y g(\alpha) d \alpha = \int_0^y \left( 1 – \frac{t \alpha}{y} \right)^{-k} d \alpha = y \int_0^1 \left( 1 – t \beta \right)^{-k} d \beta, \] by changing \( \alpha \) into \( \beta […]
Continuing our proof of Lemma A.4 and Corollary A.5 of Paper S5. It’s straightforward to check that \( g(\alpha) \) monotonically increases in \( \alpha \) strictly. The last line holds because \( g(0)=1 \) and \( g(y)=(1-t)^{-k} \). Thus, \[ (3.3) \qquad \int_0^y g(\alpha) d\alpha + 1 – \left( 1-t \right)^{-k} < U_k < […]