arXiv subjects
Publications and source records attributed to Gary Hu.
Erd\H{o}s and Graham asked whether every finite coloring of the plane must contain a monochromatic rectangle of any prescribed area. Kova\v{c} gave a negative answer by constructing a $25$-coloring with no monochromatic rectangle of area $1$. We reduce the number of colors to $21$ by replacing the square cells in his construction with regular hexagons.
A family of graphs $\mathcal{F}$ is $H$-intersecting if the edge intersection of any two graphs in $\mathcal{F}$ contains a copy of a fixed graph $H$. A fundamental problem is to determine the maximum size of such a family. The trivial lower bound of $2^{\binom{n}{2} - e(H)}$ is known to be not sharp for some graphs, such as the $P_4$ graph, as shown by Christofides. This paper presents two main contributions. First, we introduce a general construction for $H$-intersecting families based on decompositions of complete multipartite graphs, yielding new lower bounds for $H = K_{s_1, \dots, s_{k-1}, t}$. We compare this construction to a result by Balogh and Linz, showing that our bound is valid for a substantially wider range of parameters (beginning at $t \ge 2^{\sum_i s_i}$) and provides a stronger numerical bound for a large interval where both constructions are applicable. Second, we conjecture the $\frac{17}{128}$ Christofides bound for $P_4$ is optimal, which would resolve the Alon-Spencer conjecture. We computationally verify this density is optimal for families generated by connected 6-vertex host graphs with 7 or 8 edges.
A central open problem in Lie theory is the classification of finite-dimensional complex solvable Lie algebras, but classification up to isomorphism becomes increasingly difficult as the dimension grows and the number of non-isomorphic families explodes. This motivates a classification method by studying coarser spectral invariants arising from the characteristic polynomial of the adjoint representation. We prove that the characteristic polynomial is determined by the generalized weights of the adjoint action on the nilradical. This addresses the problem of expressing the spectral index in Lie-theoretic terms, gives sharp bounds in terms of the nilradical and quotient dimensions, and characterizes spectral equivalence. We then resolve a second problem concerning the higher Betti numbers of the eigenvariety complement. We endow the distinct non-$z_0$ factors of the characteristic polynomial with a natural matroid structure, which we call the spectral matroid, and use the Orlik--Solomon algebra of the associated hyperplane arrangement to determine the Betti numbers and Poincar\'{e} polynomial combinatorially and to prove log-concavity of the Betti sequence. Finally, we apply our theory to solvable Lie algebras with abelian nilradical to obtain explicit characteristic polynomials and spectral-equivalence criteria, including examples of non-isomorphic algebras with identical characteristic polynomials.
The cancellation problem asks whether $A[X_1,X_2,\ldots,X_n] \cong B[Y_1,Y_2,\ldots,Y_n]$ implies $A \cong B$. Hamann introduced the class of steadfast rings as the rings for which a version of the cancellation problem considered by Abhyankar, Eakin, and Heinzer holds. By work of Asanuma, Hamann, and Swan, steadfastness can be characterized in terms of $p$-seminormality, which is a variant of normality introduced by Swan. We prove that $p$-seminormality and steadfastness deform for reduced Noetherian local rings. We also prove that $p$-seminormality and steadfastness are stable under adjoining formal power series variables for reduced (not necessarily Noetherian) rings. Our methods also give new proofs of the facts that normality and weak normality deform, which are of independent interest.