arXiv subjects

Ruilong Liu

Publications and source records attributed to Ruilong Liu.

5 recordsLinked to original sources

arXiv2026

New Tower-Type Lower Bounds for Hypergraph Ramsey Numbers

The Ramsey number $r_k(s,m)$ is the smallest $N$ such that any red/blue coloring of the $k$-subsets of $[N]$ contains a red $s$-set or a blue $m$-set. For fixed $k$ and $s$, and for sufficiently large $m$, the tower growth rate is determined by the stepping-up lemma, but for $s=m=k+1$ the available stepping-up lemmas do not apply. Fox asked for estimates of $r_k(k+1,k+1)$. Pudl\'ak, R\"odl, and Wesley gave the first tower-type bound: $r_k(k+1,k+1)\ge s_3(\lfloor k/4\rfloor)\ge 4\operatorname{twr}_{\lfloor k/4\rfloor-4}(2)$, where $s_3(k)$ is the $3$-color shift number and $\operatorname{twr}_1(2)=2$, $\operatorname{twr}_{i+1}(2)=2^{\operatorname{twr}_i(2)}$. In this paper, for $k\ge 6$, we improve the lower bound to $r_k(k+1,k+1)> s_3\bigl(\lfloor k/2\rfloor-2\bigr)$ by overcoming an obstruction in their construction. In addition, we give an exact characterization of $s_3(k)$ and, for $k\ge 5$, obtain a new explicit lower bound $s_3(k)\ge(\operatorname{twr}_{k-2}(2))^2$, which improves the result of Pudl\'ak and R\"odl. Consequently, for $k\ge 14$, $r_k(k+1,k+1)>(\operatorname{twr}_{\lfloor k/2\rfloor-4}(2))^2$.

math.CO↗

arXiv2026

A double-exponential lower bound for $r_4(5,n)$

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. We prove that $r_4(5,n)\ge 2^{2^{cn^{1/7}}}$, where $c>0$ is an absolute constant. As a consequence, we determine the tower growth rate of $r_k(k+1,n)$, which completely solves the problem of establishing the tower growth rate for all classical off-diagonal hypergraph Ramsey numbers, first posed by Erd\H{o}s and Hajnal in 1972.

math.CO↗

arXiv2026

A Note on Generalized Erd\H{o}s-Rogers Problems

For a $k$-uniform hypergraph $F$ and positive integers $s$ and $N$, the generalized Erd\H{o}s-Rogers function $f^{(k)}_{F,s}(N)$ denotes the largest integer $m$ such that every $K_s^{(k)}$-free $k$-graph on $N$ vertices contains an $F$-free induced subgraph on $m$ vertices. In particular, if $F = K^{(k)}_t$, then we write $f^{(k)}_{t,s}(N)$ for $f^{(k)}_{F,s}(N)$. Mubayi and Suk (\emph{J. London. Math. Soc. 2018}) conjectured that $f^{(4)}_{5,6}(N)=(\log \log N)^{\Theta(1)}$. Motivated by this conjecture, we prove that $f^{(4)}_{5^{-},6}(N)=(\log\log N)^{\Theta(1)}$, where $5^{-}$ denotes the $4$-graph obtained from $K_5^{(4)}$ by deleting one edge. Our proof combines a probabilistic construction of a $2$-coloring of pairs with a stepping-up construction and an analysis of multi-layer local extremum structures. Furthermore, we derive an upper bound for a more general Erd\H{o}s-Rogers function, which implies the lower bound $r_4(6,n)\ge 2^{2^{cn^{1/2}}}$. By applying a variant of the Erd\H{o}s-Hajnal stepping-up lemma due to Mubayi and Suk, we also slightly improve the lower bound for $r_k(k+2,n)$.

math.CO↗

arXiv2026

A step towards the Erd\H{o}s-Rogers problem

For $2\le k\le t<s$, the Erd\H{o}s-Rogers function $f^{(k)}_{t,s}(N)$ denotes the largest $m$ such that every $K^{(k)}_s$-free $k$-graph on $N$ vertices contains a $K^{(k)}_t$-free induced subgraph on $m$ vertices. Mubayi and Suk (J. London Math. Soc. 2018) conjectured that $f^{(k)}_{k+1,k+2}(N)=(\log_{(k-2)}N)^{\Theta(1)}$ for $k\ge 4$, where $\log_{(i)}$ denotes the $i$-fold iterated logarithm. This is equivalent to the statement that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{\Theta(1)}$ for every $s\ge k+2$. In this paper, we introduce multi-color patterns into a random construction of a $2$-graph to build a $4$-graph, and for the first time, combine them with multi-layer extremum structures to prove that $f^{(4)}_{5,s}(N)=(\log \log N)^{\Theta(1)}$ for every $s\ge 11$. More generally, using a variant of the Erd\H{o}s-Hajnal stepping-up lemma, we also establish that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{\Theta(1)}$ for every $s\ge k+7$.

math.CO↗

arXiv2010

Quantum Secret Sharing by applying Analytic Geometry

In this paper, we investigate a novel $(2,2)$-threshold scheme and then generalize this to a $(n,n)$-threshold scheme for quantum secret sharing (QSS) which makes use of the fundamentals of Analytic Geometry. The dealer aptly selects GHZ states related to the coefficients which determine straight lines on a two-dimension plane. Then by computing each two of the lines intercept or not, we obtain a judging matrix whose rank can be used to determine the secret stored in entangled bits. Based on the database technology, authorized participants access to the database to obtain the secret information and hence the secret never appears in the channel. In this way, the eavesdroppers fail to obtain any secret by applying various attack strategies.

quant-ph↗