有向图的 Mathon 型构造与 Ramsey 数的改进下界

Degrees in oriented hypergraphs and Ramsey p-chromatic number

The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $°^+(v) \le k$ or (indegree) $°^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, int…

有向图中完美传递三角形铺砌的Ramsey-Turán型问题

Ramsey-Turán Type Problem for Perfect Transitive Triangle Tilings in Digraphs

研究有向图在独立数较小条件下，通过最小度条件保证完美传递三角形铺砌的存在性，给出了渐近最优的最小度下界。

Covering 2-colored complete digraphs by monochromatic $d$-dominating digraphs

A digraph is {\em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $d\geq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete digraph (including loops) can be covered by the vertices of at most $f(d)$ monochromatic $d$-dominating subgraphs. Note that the existence of $f(d)$ is not obvious -- indeed, the question which motivated this paper was simply to determine whether $f(d)$ is bounded, even for $d=2$. We answer this question affirmatively for all $d\geq 2$, proving $4\leq f(2)\le …

$k$次幂Paley有向图的传递子竞赛图与Ramsey数的改进下界

Transitive subtournaments of $k$-th power Paley digraphs and improved lower bounds for Ramsey numbers

本文研究$k$次幂Paley有向图中的传递子竞赛图计数，利用有限域超几何函数与Jacobi和得到三、四阶子竞赛图数量公式，并据此改进多色有向Ramsey数下界。

有序匹配的多色 Ramsey 数

Multicolor Ramsey numbers of ordered matchings

本文否定回答了有序匹配多色Ramsey数的多项式对数上界猜想，证明几乎每个完美匹配的下界指数为$q$次对数，并给出两个应用：证明有界度无环有向图的下界猜想和加强$M$-自由有序图色数下界。

Chromatic numbers of directed hypergraphs with no "bad" cycles

Imagine that you are handed a rule for determining whether a cycle in a digraph is "good" or "bad", based on which edges of the cycle are traversed in the forward direction and which edges are traversed in the backward direction. Can you then construct a digraph which avoids having any "bad" cycles, but has arbitrarily large chromatic number? We answer this question when the rule is described in terms of a finite state machine. The proof relies on Nesetril and Rodl's structural Ramsey theory of posets with a linear extension. As an application, we give a new proof of the Loop Lemma of Barto, K…

边着色与定向：应用于度数和χ-有界的应用

Edge-colouring and orientations: applications to degree- and $χ$-boundedness

论文通过推广Ramsey定理，证明大度数图中存在单色子图保持大度数，并由此得出大度数图包含大传递竞赛或大度数反向图的结论，进一步证明某些度数有界图类的性质。

Fixed-point cycles and EFX allocations

We study edge-labelings of the complete bidirected graph $\overset{\tiny\leftrightarrow}{K}_n$ with functions from the set $[d] = \{1, \dots, d\}$ to itself. We call a cycle in $\overset{\tiny\leftrightarrow}{K}_n$ a fixed-point cycle if composing the labels of its edges results in a map that has a fixed point, and we say that a labeling is fixed-point-free if no fixed-point cycle exists. For a given $d$, we ask for the largest value of $n$, denoted $R_f(d)$, for which there exists a fixed-point-free labeling of $\overset{\tiny\leftrightarrow}{K}_n$. Determining $R_f(d)$ for all $d >0$ is a na…