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Let G be a graph. We say that a hypergraph H is a Berge-G if there is a bijection ϕ : E ( G ) → E ( H ) such that e ⊆ ϕ ( e ) for all e ∈ E ( G ) . For any r-uniform hypergraph H and a real number p ≥ 1 , the p-spectral radius λ ( p ) ( H ) of H is defined as λ ( p ) ( H ) : = max x ∈ R n , ‖ x ‖ p = 1 ⁡ r ∑ { i 1 , i 2 , … , i r } ∈ E ( H ) x i 1 x i 2 ⋯ x i r . In this paper, we study the p-spectral radius of Berge-G hypergraphs. We determine the 3-uniform hypergraphs with maximum p-spectral radius for p ≥ 1 among Berge-G hypergraphs when G is a path, a cycle or a star.

In this study, a ternary salt/porous Si3N4 composite was prepared as a high-temperature shape-stabilized phase change material by pressure impregnation. In this composite, the ternary salt composed of 50 wt% sodium chloride (NaCl), 30 wt% potassium chloride (KCl), and 20 wt% magnesium chloride (MgCl2) was acted as the high-temperature phase change material. A porous Si3N4 ceramic was acted as the skeleton material and heat transfer enhancer. The porosity infiltration ratio of the pores reached 89.19%, demonstrating the good chemical compatibility of Si3N4 with chloride salts. The thermal conductivity of the composite was over six times greater than that of pure ternary salts. The thermal physical properties of the composite were stable even after 300 thermal cycles. The coefficients of thermal conductivity obtained by theoretical analysis and calculation were compared to the experimental values. All the results showed that this composite is a promising candidate as a shape-stabilized high-temperature phase change heat storage material.

For $r\geq 3$, let $f_r\colon [0,\infty)\to [1,\infty)$ be the unique analytic function such that $f_r({k\choose r})={k-1\choose r-1}$ for any $k\geq r-1$. We prove that the spectral radius of an $r$-uniform hypergraph $H$ with $e$ edges is at most $f_r(e)$. The equality holds if and only if $e={k\choose r}$ for some positive integer $k$ and $H$ is the union of a complete $r$-uniform hypergraph $K_k^r$ and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs., 16 pages

For a fixed set of positive integers R , we say H is an R -uniform hypergraph, or R -graph, if the cardinality of each edge belongs to R . For a graph G = ( V , E ) , a hypergraph H is called a Berge- G , if there is an injection i : V ( G ) → V ( H ) and a bijection f : E ( G ) → E ( H ) such that for all e = u v ∈ E ( G ) , we have { i ( u ) , i ( v ) } ⊆ f ( e ) . We define the cover Turan number of a graph G , denoted as ex ˆ R ( n , B G ) , as the maximum number of edges in the 2-shadow of an n -vertex Berge- G -free R -graph, where the 2-shadow H of a hypergraph H is a graph such that an edge e ∈ E ( H ) if and only if e ⊆ h for some h ∈ E ( H ) . In this paper, we show an Erdős–Stone–Simonovits-type upper bound on the cover Turan number of graphs and determine the cover Turan density of all graphs when the host hypergraph is 3-uniform.

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4] . Let U m , d denote the uniform matroid of rank d on a set of m + d elements. Gedeon et al. (2017) [7] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of U m , d using equivariant Kazhdan-Lusztig polynomials. In this paper we give an alternative explicit formula, which allows us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of U m , d for 2 ≤ m ≤ 15 and all d's. The case m = 1 was previously proved by Gedeon et al. (2017) [8] . We further determine the Z-polynomials of all U m , d 's and prove the real-rootedness of the Z-polynomials of U m , d for 2 ≤ m ≤ 15 and all d's. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Young's formula for the Kazhdan-Lusztig polynomials of U m , d 's without using the equivariant Kazhdan-Lusztig polynomials.

Let $2^{[n]}$ denote the power set of $[n]:=\{1,2,..., n\}$. A collection $\B\subset 2^{[n]}$ forms a $d$-dimensional {\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \subseteq [n]$, all non-empty with perhaps the exception of $X_0$, so that $\B={X_0\cup \bigcup_{i\in I} X_i\colon I\subseteq [d]}$. Let $b(n,d)$ be the maximum cardinality of a family $\F\subset 2^X$ that does not contain a $d$-dimensional Boolean algebra. Gunderson, R\"odl, and Sidorenko proved that $b(n,d) \leq c_d n^{-1/2^d} \cdot 2^n$ where $c_d= 10^d 2^{-2^{1-d}}d^{d-2^{-d}}$. In this paper, we use the Lubell function as a new measurement for large families instead of cardinality. The Lubell value of a family of sets $\F$ with $\F\subseteq \tsupn$ is defined by $h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}$. We prove the following Tur\'an type theorem. If $\F\subseteq 2^{[n]}$ contains no $d$-dimensional Boolean algebra, then $h_n(\F)\leq 2(n+1)^{1-2^{1-d}}$ for sufficiently large $n$. This results implies $b(n,d) \leq C n^{-1/2^d} \cdot 2^n$, where $C$ is an absolute constant independent of $n$ and $d$. As a consequence, we improve several Ramsey-type bounds on Boolean algebras. We also prove a canonical Ramsey theorem for Boolean algebras., Comment: 10 pages

The crown $\Oh_{2t}$ is a height-2 poset whose Hasse diagram is a cycle of length $2t$. A family $\F$ of subsets of $[n]:=\{1,2..., n\}$ is {\em $\Oh_{2t}$-free} if $\Oh_{2t}$ is not a weak subposet of $(\F,\subseteq)$. Let $\La(n,\Oh_{2t})$ be the largest size of $\Oh_{2t}$-free families of subsets of $[n]$. De Bonis-Katona-Swanepoel proved $\La(n,\Oh_{4})= {n\choose \lfloor \frac{n}{2} \rfloor} + {n\choose \lceil \frac{n}{2} \rceil}$. Griggs and Lu proved that $\La(n,\Oh_{2t})=(1+o(1))\nchn$ for all even $t\ge 4$. In this paper, we prove $\La(n,\Oh_{2t})=(1+o(1))\nchn$ for all odd $t\geq 7$., 17 pages

The spectral radius ρ ( G ) of a graph G is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph G with ρ ( G ) ⩽ 3 2 2 is either a dagger, an open quipu, or a closed quipu. The reverse statement is not true. Many open quipus and closed quipus have spectral radii greater than 3 2 2 . In this paper we proved the following results. For any open quipu G on n vertices ( n ⩾ 6 ) with spectral radius less than 3 2 2 , its diameter D ( G ) satisfies D ( G ) ⩾ ( 2 n - 4 ) / 3 . This bound is tight. For any closed quipu G on n vertices ( n ⩾ 13 ) with spectral radius less than 3 2 2 , its diameter D ( G ) satisfies n 3 D ( G ) ⩽ 2 n - 2 3 . The upper bound is tight while the lower bound is asymptotically tight. Let G n , D min be a graph with minimal spectral radius among all connected graphs on n vertices with diameter D . For n ⩾ 14 and D ∈ [ n 2 , 2 n - 5 3 ] , we proved that G n , D min is the unique graph obtained by attaching two paths of lengths D - ⌊ n 2 ⌋ and D - ⌈ n 2 ⌉ to a pair of antipodal vertices of the even cycle C 2 ( n - D ) . This result is tight. Thus we settled a conjecture of Cioabaˇ–van Dam–Koolen–Lee, who previously proved a special case D = n + e 2 for e ∈ { 1 , 2 , 3 , 4 } and n large enough.

Let G be a triangle-free graph with maximum degree at most 3. Staton proved that the independence number of G is at least 5 14 | V ( G ) | . Heckman and Thomas conjectured that Staton's result can be strengthened into a bound on the fractional chromatic number of G , namely ? f ( G ) ? 14 5 . Recently, Hatami and Zhu proved that ? f ( G ) ? 3 - 3 64 . In this paper, we prove ? f ( G ) ? 3 - 3 43 .

This paper is motivated by a recent result of Wolf on the minimum number of monochromatic 4-term arithmetic progressions (4-APs, for short) in Z"p, where p is a prime number. Wolf proved that there is a 2-coloring of Z"p with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic. In this paper, we present an explicit and simple construction of a 2-coloring with 9.3% fewer monochromatic 4-APs than random 2-colorings. This problem leads us to consider the minimum number of monochromatic 4-APs in Z"n for general n. We obtain both lower bound and upper bound on the minimum number of monochromatic 4-APs in Z"n. Wolf proved that any 2-coloring of Z"p has at least (1/16+o(1))p^2 monochromatic 4-APs. We improve this lower bound to (7/96+o(1))p^2. Our method for Z"n naturally apply to the similar problem on [n]. In 2008, Parillo, Robertson, and Saracino constructed a 2-coloring of [n] with 14.6% fewer monochromatic 3-APs than random 2-colorings. In 2010, Butler, Costello, and Graham used a new method to construct a 2-coloring of [n] with 17.35% fewer monochromatic 4-APs (and 26.8% fewer monochromatic 5-APs) than random 2-colorings. Our construction gives a 2-coloring of [n] with 33.33% fewer monochromatic 4-APs (and 57.89% fewer monochromatic 5-APs) than random 2-colorings.

We consider the diameter of a random graph G(n,p) for various ranges of p close to the phase transition point for connectivity. For a disconnected graph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of random graph G(n,p) is close to lognlog(np) if np->~. Moreover if nplogn=c>8, then the diameter of G(n,p) is concentrated on two values. In general, if nplogn=c>c"0, the diameter is concentrated on at most [email protected]?1/c"[email protected]?+4 values. We also proved that the diameter of G(n,p) is almost surely equal to the diameter of its giant component if np>3.6.

Let tr(n, r+1) denote the smallest integer m such that every r-uniform hypergraph on n vertices with m+1 edges must contain a complete graph on r+1 vertices. In this paper, we prove thatlimn→∞t3(n, 4)(n3)⩽3+1712=0.593592….