Combinatorial Game Distributions of Steiner Systems

The $P$-position sets of some combinatorial games have special combinatorial structures. For example, the $P$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, denoted by $D_{\text{sh}}$. However, few games were known to be related to Steiner systems in this way. For a given Steiner system, we construct a game whose $P$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $D_{\text{sh}}$ among the 5040 isomorphic $S(5, 6, 12)$ with point set $\{0, 1, ..., 11\}$. For each $S(5, 6, 12)$, our construction produces a game whose $P$-position set is its block set. From $D_{\text{sh}}$, we obtain the hexad game, and this game is characterized as the unique game with the minimum number of positions among the obtained 5040 games. Second, we characterize projective Steiner triple systems by using game distributions. Here, the game distribution of a Steiner system $D$ is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to $D$. We find that the game distribution of an $S(t, t + 1, v)$ can be decomposed into symmetric components and that a Steiner triple system is projective if and only if its game distribution has a unique symmetric component.

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v ≡ 39 (mod 72), or $$v \equiv 4^e48 + 3$$ v ≡ 4 e 48 + 3 (mod $$4^e96$$ 4 e 96 ) and $$e \ge 0$$ e ≥ 0 , there exists a KTS on v points having at least $$v-3$$ v - 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

On deficiency problems for graphs

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$ , the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer t such that the join $G*K_t$ has property $\mathcal P$ . In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an n-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$ -factor (for any fixed $r\geq 3$ ). In this paper, we resolve their problem fully. We also give an analogous result that forces $G*K_t$ to contain any fixed bipartite $(n+t)$ -vertex graph of bounded degree and small bandwidth.

On the Intersection Distribution of Degree Three Polynomials and Related Topics

The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed by Li and Pott [\emph{Finite Fields and Their Applications, 66 (2020)}], which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in\mathbb{F}_q\}$. The intersection distribution has an underlying geometric interpretation, which indicates the intersection pattern between the graph of $f$ and the lines in the affine plane $AG(2,q)$. When $q$ is even, the long-standing open problem of classifying o-polynomials can be rephrased in a simple way, namely, classifying all polynomials which have the same intersection distribution as $x^2$. Inspired by this connection, we proceed to consider the next simplest case and derive the intersection distribution for all degree three polynomials over $\mathbb{F}_q$ with $q$ both odd and even. Moreover, we initiate to classify all monomials having the same intersection distribution as $x^3$, where some characterizations of such monomials are obtained and a conjecture is proposed. In addition, two applications of the intersection distributions of degree three polynomials are presented. The first one is the construction of nonisomorphic Steiner triple systems and the second one produces infinite families of Kakeya sets in affine planes with previously unknown sizes.