arXiv subjects

Yaping Mao

Publications and source records attributed to Yaping Mao.

At least 19 recordsLinked to original sources

arXiv2026

Multicolor Ramsey and list Ramsey numbers for star-like trees

For a graph $H$, the $k$-color Ramsey number $r(H;k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $H$. A $k$-list assignment has $|L(e)|=k$ for every edge. The list Ramsey number $r_\ell(H;k)$ is the least integer $N$ for which there exists a $k$-list assignment on $E(K_N)$ such that every coloring from the lists contains a monochromatic copy of $H$. Let $K_{1,n}$ be a star, $S(n,m)$ the double star obtained by joining the centers of $K_{1,n}$ and $K_{1,m}$, and $S_n^m$ the graph obtained from $K_{1,n}$ by subdividing $m$ edges once. Alon et al.\ conjectured that $r_\ell(K_{1,n};k)=r(K_{1,n};k)$ for all $k,n\ge1$. In this paper, we confirm their conjecture for all $k\ge1$ and $n\ge3$ by a unified direct proof. For even $k\ge4$ and under explicit parameter conditions, we prove that $r(S(n,m);k)=kn+m+2$ for even $n$ and $r(S_n^m;k)=k(n-1)+m+2$ for odd $n$. For two colors, we establish a general list Ramsey lower bound and determine the common values of Ramsey and list Ramsey numbers for double and subdivided stars in explicit parameter ranges. These results close several gaps in the known bounds.

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arXiv2026

Dense ascending waves: A resolution of the Alon-Spencer conjecture

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. A strictly increasing sequence of integers $x_1<\cdots<x_k$ is an \emph{ascending wave} if its consecutive differences are nondecreasing. Let $g(n)$ be the largest integer $k$ such that every set $A\subseteq[n]$ with $|A|\ge n/2$ contains an ascending wave of length $k$. Alon and Spencer proved that \[ c_1\frac{(\log n)^2}{\log\log n}\le g(n)\le c_2(\log n)^2 \] for all sufficiently large $n$, and they conjectured that the factor $\log\log n$ in the lower bound can be removed. In this paper, we confirm their conjecture.

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arXiv2026

Gap spectra and densities of slow Fibonacci walks

Let $F_1=F_2=1$ and $F_{t+2}=F_{t+1}+F_t$ for $t\geq1$. For every $n\geq2$, there are unique integers $a,b,t$ such that $n=aF_t+bF_{t-1}$ with $t\geq2$ and $1\leq a\leq b\leq F_t$. The Fibonacci walk with initial pair $(b,a)$ reaches $n$ as late as possible, and the term following $n$ in this walk is $\lfloor\phi n\rfloor$ when $t$ is even and $\lceil\phi n\rceil$ when $t$ is odd, where $\phi=(1+\sqrt5)/2$. Let $D=\{d_1<d_2<\cdots\}$ and $U=\{u_1<u_2<\cdots\}$ be the sets corresponding to even and odd $t$, respectively. For $\ell,m\geq1$, define $D_\ell=\{d_{k+\ell}-d_k:k\geq1\}$, $U_\ell=\{u_{k+\ell}-u_k:k\geq1\}$, $D_\ell(m)=\{d_k:d_{k+\ell}-d_k=m\}$ and $U_\ell(m)=\{u_k:u_{k+\ell}-u_k=m\}$. Chung, Graham and Spiro conjectured that $D_\ell=U_\ell$ for all $\ell$, and asked for the densities of $D_\ell(m)$ and $U_\ell(m)$, especially when $\ell=1$. In this paper, we determine the third and fourth order gap spectra, and show that the conjecture holds for $\ell=3$ but fails for $\ell=4$. We also answer their density question by characterizing when $D_\ell(m)$ and $U_\ell(m)$ have natural densities and proving that their logarithmic densities always exist and are equal. For $\ell=1$, we give the exact logarithmic densities.

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arXiv2026

New upper bound for multicolor Ramsey numbers

Let $R_r(k)$ denote the diagonal $r$-color graph Ramsey number. We prove that there exist absolute constants $c,K>0$ such that \[ R_r(k)\le \exp\!\left(-c\frac{k}{r^2\log^4(2r)}\right)r^{rk} \] for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. The proof combines a positive-coefficient root filter of variable order with a retained-spine refinement of the multicolor book method.b

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arXiv2026

Ramsey multiplicity for ordered graphs

Let $\cG_1,\ldots,\cG_k$ be fixed vertex-ordered graphs, each containing at least one edge. The ordered Ramsey number $\oR(\cG_1,\ldots,\cG_k)$ is the least integer $N$ such that every $k$-edge-coloring of the ordered complete graph $\cK_N$ contains an order-preserving copy of $\cG_i$ in color $i$ for some $i\in[k]$. For positive weights $\blambda=(\lambda_1,\ldots,\lambda_k)$, let $\oM_{\blambda}(n;\cG_1,\ldots,\cG_k)$ denote the minimum weighted number of correctly colored, order-preserving copies of the target graphs over all $k$-edge-colorings of $\cK_n$. When $\blambda=\bf{1}$, $\oM_{\bf{1}}(n;\cG_1,\ldots,\cG_k)=\oM(n;\cG_1,\ldots,\cG_k)$ is called the ordered Ramsey multiplicity. In this paper, we first establish the amplification inequality \[ \oM_{\blambda}(n;\cG_1,\ldots,\cG_k) \ge \oM_{\blambda}(t;\cG_1,\ldots,\cG_k) \frac{\binom{n}{\hmin}}{\binom{t}{\hmin}}, \] where $h_i=v(\cG_i),\hmin=\min_{i\in[k]}h_i$, and $n\ge t\ge\oR(\cG_1,\ldots,\cG_k)$. Let $\cS_{r,s}$ be the ordered star whose center has $r-1$ leaves to its left and $s-1$ leaves to its right, and let $\bB_m$ be the family of all ordered perfect matchings on $[2m]$ containing the edge $\{1,2m\}$. We apply the amplification inequality to obtain the multiplicity lower bounds for ordered stars and ordered perfect matchings. We then obtain the upper bound $\oM_{\boldsymbol\lambda} (n;\cS_{r_1,s_1},\cS_{r_2,s_2}) \le \min\{\lambda_1 B_{h_1}(n),\lambda_2 B_{h_2}(n)\}$ by constructions, where $B_{h_i}(n):= \binom{\lfloor n/2\rfloor}{h_i} + \binom{\lceil n/2\rceil}{h_i}$ and $h_i=r_i+s_i-1$ for $i\in [2]$. We also derive a random-coloring upper bound for ordered stars and prove \[\oM(n; \bB_m,\bB_m) \le \binom{n}{2m} \frac{(2m-2)!}{2^{2m-2}(m-1)!}.\] Finally, we establish a regularity-based lifting theorem for ordered colorings.

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arXiv2026

The inversion number of a path-reversed tournament: Resolving a conjecture of Belkhechine, Bouaziz, Boudabbous, and Pouzet

Let $D$ be a tournament and let $X\subseteq V(D)$. The inversion of $X$ reverses all arcs whose both endpoints lie in $X$ and leaves every other arc unchanged. A family of inversions is a decycling family if applying all of them produces an acyclic, equivalently transitive, tournament. The inversion number $\inv(D)$ is the minimum size of such a family. Let $Q_n$ be the tournament on $[n]$ obtained from the natural transitive tournament by reversing precisely the consecutive pairs $12,23,\ldots,(n-1)n$. Belkhechine, Bouaziz, Boudabbous, and Pouzet conjectured in their unpublished manuscript that a natural path-reversed family has inversion number exactly $\left\lfloor(n-1)/2\right\rfloor$. The same problem was later recorded by Bang-Jensen, da Silva, and Havet and by Alon, Powierski, Savery, Scott, and Wilmer. In this paper we resolve this conjecture.

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arXiv2026

Solution to a conjecture of Alon, D\k{e}bski, Grytczuk and Przyby\l{}o on fixed-cardinality arithmetic progressions

Fix a positive integer $n$, and put $B_d=\{d,2d,\ldots,nd\}$. Let $M_k(n)$ be the least integer $m$ for which one translate of each of $B_1,\ldots,B_k$ can be placed pairwise disjointly in $[m]$. We prove that, for every $\eps\in(0,1)$ and all sufficiently large $k$, one has $M_k(n)\le n\lceil(1+\eps)k\rceil$. Since the trivial counting bound gives $M_k(n)\ge nk$, it follows that $M_k(n)=(1+o(1))nk$ for every fixed $n$. This confirms a conjecture of Alon, D\k{e}bski, Grytczuk and Przyby\l{}o on prescribed-difference packings of fixed-cardinality arithmetic progressions.

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arXiv2026

Multiplicity for partially ordered sets

Let $\mathcal Q=\{Q_a:a\geq1\}$ be a nested family of finite posets such that $Q_a\subseteq Q_{a+1}$ and $|Q_a|<|Q_{a+1}|$. For a poset $Q$, let $\mathcal C_t(Q)$ denote the set of all strict $t$-chains in $Q$. Given an $r$-coloring of $\mathcal C_t(Q_a)$ and posets $P_1,\ldots,P_r$, a weak copy of $P_i$ is called monochromatic of color $i$ if all $t$-chains in the copy have color $i$; the strong version is defined in the same way for induced copies. The corresponding weak and strong multiplicity parameters are the minimum possible total number of such monochromatic copies in the host poset.For the Boolean lattice $B_n$, define $E_n={(S,T,U)\in B_n^3:S\subsetneq T\subsetneq U,\ |S|+|T|=|U|}.$ For a two-coloring $\chi:B_n\to{0,1}$, a triple $(S,T,U)\in E_n$ is monochromatic if $\chi(S)=\chi(T)=\chi(U)$. Let $R^{\mathrm{arith}}_2$ be the least integer $n$ such that every two-coloring of $B_n$ contains a monochromatic triple in $E_n$, and let $M^{\mathrm{arith}}_2(B_n)$ be the minimum number of monochromatic triples in $E_n$ over all two-colorings of $B_n$. We prove that $R^{\mathrm{arith}}_2=9.$ Moreover, $|E_n|=\binom{2n}{n}-[x^n](1+x+x^2)^n-2^n+1=\frac{4^n}{\sqrt{\pi n}}\bigl(1+o(1)\bigr),$ and $2^{\delta n+o(n)}\le M^{\mathrm{arith}}_2(B_n)\le 2^{\gamma n+o(n)}, $ where $\delta\approx 1.356779$ and $\gamma\approx 1.567837$ are explicit entropy constants. For general nested host families, we prove a double-counting lower bound for strong poset multiplicity. For an arbitrary finite host poset $R$, we also introduce a Fourier-M\"obius method and give an exact Fourier expansion for strong multiplicity, a Parseval-type error bound, and a spectral lower bound.

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arXiv2026

A complete solution to the biased Alon-Krivelevich-Spencer-Szab\'o criterion problem for the discrepancy game

Let $H=(V,\mathcal E)$ be a finite hypergraph. For positive integers $p$ and $q$, the $(p:q)$-biased discrepancy game on $H$ is played in complete rounds. In each round, Balancer first claims $p$ previously unclaimed vertices, and then Unbalancer claims $q$ previously unclaimed vertices. Let $B$ and $U$ be the final sets of vertices claimed by Balancer and Unbalancer, respectively. For an edge $e\in\mathcal E$, define $D_e = q|B\cap e|-p|U\cap e| = (p+q)|B\cap e|-p|e|$. Thus $D_e$ measures the deviation of Balancer's share of $e$ from the density $p/(p+q)$. In 2005, Alon, Krivelevich, Spencer and Szab\'o proved a Chernoff-type potential criterion for the unbiased alternating discrepancy game, corresponding to the case $p=q=1$, and asked for a biased analogue for general $p,q$. In this paper, we prove a complete biased analogue in the complete-round formulation. More precisely, for every finite hypergraph $H=(V,\mathcal E)$ and every fixed bias $(p:q)$, we give an explicit exponential condition under which Balancer has a strategy forcing $-L_e^- \le D_e \le L_e^+$ for every $e\in\mathcal E$, where $L_e^+$ and $L_e^-$ are prescribed edge-dependent target values.

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arXiv2026

Ramsey-Tur\'{a}n theory for partially-ordered sets

We introduce weak and strong poset Ramsey-Tur\'an numbers for $t$-chains in host poset families, focusing on the Boolean lattice family $\mathcal{B}=\{B_n:n\ge 1\}$. For any poset $P$, we show $\operatorname{RT}(\mathcal{B};n,P,l,t)\le \operatorname{RT}^{\sharp}(\mathcal{B};n,P,l,t)$, with equality when $P$ is a chain. In particular, for $t=1$, $\operatorname{RT}(\mathcal{B};n,C_k,l)=\operatorname{RT}^{\sharp}(\mathcal{B};n,C_k,l)=(k-1)(l-1)$. We also give universal upper bounds for both versions. For fixed $k,l,t$ with $\min\{l-1,k-1\}\ge 1$, we prove $\operatorname{RT}^{\sharp}(\mathcal{B};n,A_k,l,t)=\Theta(n^t)$. More generally, for every non-chain poset $P$, the strong number is $\Theta(n^t)$ for fixed $l,t$. Finally, if $h(P)=r>t$ and $l(n)=\lfloor M_n^\beta\rfloor$ with $0<\beta\le \alpha<1$, then both weak and strong versions admit lower bounds of order $\Omega\!\left(2^{\beta n}n^{-\beta/2}\right)$.

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arXiv2026

Diagonal Ramsey numbers for wheels

The Ramsey number $\mathrm{R}(G_1,G_2)$ is the smallest integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ contains either a red copy of $G_1$ or a blue copy of $G_2$. In 2022, the third author and others gave lower and upper bounds of the Ramsey number $\mathrm{R}(W_n,W_n)$, where $W_n$ is the wheel graph with $n$ vertices. In this paper, we improve their bounds by showing that $3n-2\leq \mathrm{R}(W_n,W_n)\leq 6n-6$ for even $n\geq 8$ and $2n\leq \mathrm{R}(W_n,W_n)\leq \frac{9n-7}{2}$ for odd $n\geq 7$. Furthermore, we give recursive bounds for the $k$-colored Ramsey number for $W_n$.

math.CO↗

arXiv2026

From Halin's Edge Removability to Matching Removability in $k$-Connected Graphs

We study matching-removability under the degree/connectivity regime of Halin's theorem, which asserts that every $k$-connected graph $G$ with minimum degree $\delta(G)\ge k+1$ contains an edge $e$ such that $G-e$ remains $k$-connected. For $k,\ell\ge 1$, an $\ell$-matching is a matching of size $\ell$. A matching $M$ in a $k$-connected graph $G$ is {\it $k$-removable} if $G-M$ remains $k$-connected. We improve Halin's result by proving that every $k$-connected graph $G$ with $\delta(G)\ge k+1$ contains a $k$-removable $2$-matching, except when $k=1$ and $G$ is a cycle. For small $k$ we obtain stronger bounds: (i) $k=1$: a 1-removable $\min\{\lfloor n/2\rfloor,\delta(G)\}$-matching; (ii) $k=2$: a 2-removable $\lceil(\delta(G)+1)/2\rceil$-matching, with a unique tight exception when $\delta(G)$ is even and $G\cong K_{\delta(G)+1}$; and (iii) $k=3$: for $\delta(G)\ge 5$, a $3$-removable $\lceil(\delta(G)+1)/2\rceil$-matching. All these bounds are optimal with respect to removable matching size and minimum degree. We also show that for every $n\ge 2\delta$, there exists a $k$-connected $n$-vertex graph $G$ with minimum degree $\delta$ that does not contain a $k$-removable matching of size at least $\delta(G)+1$. Moreover, for $k\le 2$ there exists a $k$-removable $(\delta(G)-c)$-matching for some $c\le 3$, which is optimal up to the additive constant.

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arXiv2026

Diameter-Ramsey triangles below the $135^\circ$

A finite Euclidean set is diameter-Ramsey if, for every number of colors, some finite same-diameter witness has the property that every coloring of the witness contains a monochromatic congruent copy of the set. Frankl, Pach, Reiher and R\"odl asked whether any obtuse triangle is diameter-Ramsey. We prove the stronger statement that every non-degenerate triangle whose largest angle is strictly smaller than $135^\circ$ is diameter-Ramsey. Together with the theorem of Corsten and Frankl that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, this gives the sharp classification for the two open angular ranges on either side of $135^\circ$. The proof uses a weighted $k$-subset configuration with non-negative coefficients; a finite binary-tree construction realizes the required two prescribed overlaps, and the ordinary hypergraph Ramsey theorem then forces a monochromatic copy of the triangle.

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arXiv2026

Counterexamples to the Corsten-Frankl conjecture on diameter-Ramsey simplices

Corsten and Frankl conjectured that a simplex is diameter-Ramsey if and only if its circumcenter lies in its convex hull. We disprove this conjecture in every dimension $d\ge 3$. The main tool is a sufficient criterion based on a higher-order deficit decomposition: if the squared deficits $D^2-\|p_i-p_j\|^2$ admit a nonnegative decomposition over subsets of the vertex set, with total mass at most $D^2$, then the simplex is diameter-Ramsey. The pairwise deficit criterion of Frankl--Pach--Reiher--R\"odl is recovered as a special case. As an application, for every $d\ge 3$ we construct a diameter-Ramsey $d$-simplex whose circumcenter lies outside its convex hull. A particularly simple family has squared edge lengths $\|p_1-p_2\|^2=\|p_1-p_j\|^2=7~ (4\le j\le d+1)$, $\|p_1-p_3\|^2=4$, and $\|p_i-p_j\|^2=4 ~ (2\le i<j\le d+1)$.

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arXiv2026

Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erd\H{o}s-Faudree-Schelp conjectures

For finite graphs $G$ and $H$, let $\RR(G,H)$ denote the isomorphism classes of Ramsey-minimal graphs for $(G,H)$. We prove two 1981 conjectures of Burr, Erd\H{o}s, Faudree, Rousseau, and Schelp: Ramsey-finiteness is preserved by adjoining disjoint matchings, and $(G,H)$ is Ramsey-infinite unless both graphs are odd stars or one graph has a $K_2$ component. We also replace Burr's stronger 1979 survey characterization by the correct necessary-and-sufficient form: apart from the matching case and the odd-star-with-matchings case, the only additional finite pairs are Faudree's star-forest family.

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arXiv2026

The Erd\H{o}s-Faudree Problems and the Isolate-Free Core

In 1981, Erd\H{o}s and Faudree asked whether there exists an infinite family of graphs $G_N$ on $N$ vertices with $\Delta(G_N)<N-1$ and $\sri(G_N)=1$, and whether every family with $|V(G_N)|=N$ and $\Delta(G_N)<c$ for some fixed constant $c$ must satisfy $\sri(G_N)\to 0$. We show first that the literal forms of the two questions are controlled entirely by isolated vertices: for every nonempty graph $G$, the whole sequence $\bigl(\sr(tK_2,G)\bigr)_{t\ge 1}$ depends only on the isolate-free core $\core(G)$. Consequently, Problem 1 has a positive answer and Problem~2 has a negative answer in exactly their original form. We then turn to the genuine content behind the two problems. For Problem 1 we study connected graphs and prove a complete limit theorem: for every $\alpha\in[0,1]$ there exists a family of connected bipartite graphs $G_N$ with $|V(G_N)|=N$ and $\sri(G_N)\to\alpha$; in particular there are connected graphs with $\Delta(G_N)=N-2$ and $\sri(G_N)\to 1$. For Problem~2 we prove a strengthened positive statement: if $\Delta(G_N)<c$ for a fixed constant $c$ and the isolate-free core of $G_N$ has order tending to infinity, then $\sri(G_N)\to 0$. In particular every connected bounded-degree family satisfies $\sri(G_N)\to 0$. Thus the original Erd\H{o}s-Faudree questions are resolved in their literal form, and the mechanism behind their connected and disconnected behavior is identified precisely.

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arXiv2026

On two conjectures for generalized off-diagonal Schur numbers

For an integer $t \geq 3$, let $\mathcal{L}(t)$ denote the linear equation $x_1 + x_2 + \cdots + x_{t-1} = x_t,$ where all variables are positive integers. For integers $k \geq 1$ and $t_0,t_1,\dots,t_{k-1} \geq 3$, the generalized Schur number $S(k;t_0,t_1,\dots,t_{k-1})$ is the least positive integer $N$ such that every $k$-coloring of $[1,N]$, for some $i \in \{0,1,\dots,k-1\}$, a solution to $\mathcal{L}(t_i)$ with all variables monochromatic in color $i$. In 2015, Ahmed and Schaal proposed a conjecture: $S(3 ; 3, t, u)>3 t u-t u-u-1$ for $3=t<u$ and $3<t \leq u$. In this paper, we confirm this conjecture. At the same paper, they also conjecture that $S(3 ; s, t, u)=s t u-t u-u-1$ for $4 \leq s \leq t \leq u$. Motivated by the second conjecture, we give a recursive lower bound of $S(r; k_0, k_1, \dots, k_{r-1})$ and upper bounds for $S(r; k_0, k_1, \dots, k_{r-1})$ and $S(r;k_0,\dots,k_{r-2},u)$ for all sufficiently large $u$.

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arXiv2026

Erd\H{o}s-Gy\'{a}rf\'{a}s problem for partially ordered sets

Given integers $p,q,t$ with $1 \le t \le p$ and $1 \le q \le h_p(t)$, a strong $(p,q,t)$-coloring of the Boolean lattice $B_n$ is a coloring of its $t$-chains such that every induced copy of $B_p$ in $B_n$ uses at least $q$ colors on its $t$-chains. Let $f_t^{\sharp}(n,p,q)$ denote the minimum number of colors in such a coloring. We study this Boolean-lattice analogue of the Erd\H{o}s-Gy\'{a}rf\'{a}s function.We first show that every finite poset strongly embeds into a Boolean lattice. Combined with a structural Ramsey theorem for finite posets with linear extensions, this implies the existence of the strong Boolean Ramsey number $\mathrm{R}^{\sharp}_{k,t}(\mathcal{B}\mid Q)$ for every integer $k\ge1$, every $t\ge1$, and every nonempty finite poset $Q$. In particular, this gives an affirmative answer to a problem of Cox and Stolee and yields the existence of $f_t^{\sharp}(n,p,2)$. Next, using the symmetric Lov\'asz local lemma, we obtain a probabilistic upper bound on $f_t^{\sharp}(n,p,q)$. Finally, we prove lower bounds by combining Tur\'an-type extremal estimates for $t$-chains, a double-counting argument, and a generalized Lubell-type framework for $t$-chains.

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