Monochromatic Sets Research Articles

Owings-like theorems for infinitely many colours or finite monochromatic sets

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers κ,θ, every colouring c:G⟶θ yields a subset X⊆G with |X|=κ such that X+X is monochromatic. (Owings's problem asks this for G=Z, θ=2 and κ=ℵ0; this is known to be false for the same G and κ but θ=3.) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.

Monochromatic products and sums in the rationals

Abstract We show that every finite coloring of the rationals contains monochromatic sets of the form $\{x,y,xy,x+y\}$ .

A counterexample to a conjecture on the chromatic number of r $r$‐stable Kneser hypergraphs

AbstractLet be a set of subsets of and be an integer. The ‐colorability defect of is the least number of elements of that need to be removed such that the remaining ground set can be ‐colored without inducing monochromatic sets in . The set is a subset of containing all elements of such that any two members of are at least at distance apart on the ‐cycle. The main purpose of this note is to give a counterexample to a conjecture raised by Florian Frick, which says if is a set of subsets of and , then the number of blocks needed to partition the elements of such that no pairwise disjoint sets lie in the same block is at least .

The ultrafilter number and

AbstractThe cardinal invariant $\mathfrak {hm}$ is defined as the minimum size of a family of $\mathsf {c}_{\mathsf {min}}$ -monochromatic sets that cover $2^{\omega }$ (where $\mathsf {c}_{\mathsf {min}}( x,y) $ is the parity of the biggest initial segment both x and y have in common). We prove that $\mathfrak {hm}=\omega _{1}$ holds in Shelah’s model of $\mathfrak {i<u},$ so the inequality $\mathfrak {hm<u}$ is consistent with the axioms of $\mathsf {ZFC}$ . This answers a question of Thilo Weinert. We prove that the diamond principle $\mathfrak {\Diamond }_{\mathfrak {d}}$ also holds in that model.

On the Chromatic Numbers Corresponding to Exponentially Ramsey Sets

In this paper, nontrivial upper bounds on the chromatic numbers of the spaces $$ {\mathrm{\mathbb{R}}}_p^n=\left({\mathrm{\mathbb{R}}}^n, lp\right) $$ with forbidden monochromatic sets are proved. In the case of a forbidden rectangular parallelepiped or a regular simplex, explicit exponential lower bounds on the chromatic numbers are obtained. Exact values of the chromatic numbers of the spaces $$ {\mathrm{\mathbb{R}}}_p^n $$ with a forbidden regular simplex in the case p = ∞ are found. Bibliography: 39 titles.

Hindman-like theorems with uncountably many colours and finite monochromatic sets

A particular case of the Hindman–Galvin–Glazer theorem states that, for every partition of an infinite abelian group G G into two cells, there will be an infinite X ⊆ G X\subseteq G such that the set of its finite sums { x 1 + ⋯ + x n ∣ n ∈ N ∧ x 1 , … , x n ∈ X are distinct } \{x_1+\cdots +x_n \mid n\in \mathbb N\wedge x_1,\ldots ,x_n\in X\text { are distinct}\} is monochromatic. It is known that the same statement is false, in a very strong sense, if one attempts to obtain an uncountable (rather than just infinite) X X . On the other hand, a recent result of Komjáth states that, for partitions into uncountably many cells, it is possible to obtain monochromatic sets of the form F S ( X ) \mathrm {FS}(X) , for X X of some prescribed finite size, when working with sufficiently large Boolean groups. In this paper, we provide a generalization of Komjáth’s result, and we show that, in a sense, this generalization is the strongest possible.

Efficacy of single-source rapid kV-switching dual-energy CT for characterization of non-uric acid renal stones: a prospective ex vivo study using anthropomorphic phantom.

To investigate the accuracy of rapid kV-switching single-source dual-energy computed tomography (rsDECT) for prediction of classes of non-uric-acid stones. Non-uric-acid renal stones retrieved via percutaneous nephrolithotomy were prospectively collected between January 2017 and February 2018 in a single institution. Only stones ≥ 5mm and with pure composition (i.e., ≥ 80% composed of one component) were included. Stone composition was determined using Fourier Transform Infrared Spectroscopy. The stones were scanned in 32-cm-wide anthropomorphic whole-body phantom using rsDECT. The effective atomic number (Zeff), the attenuation at 40keV (HU40), 70keV (HU70), and 140keV (HU140) virtual monochromatic sets of images as well as the ratios between the attenuations were calculated. Values of stone classes were compared using ANOVA and Mann-Whitney U test. Receiver operating curves and area under curve (AUC) were calculated. A p value < 0.05 was considered statistically significant. The final study sample included 31 stones from 31 patients consisting of 25 (81%) calcium-based, 4 (13%) cystine, and 2 (6%) struvite pure stones. The mean size of the stones was 9.9 ± 2.4mm. The mean Zeff of the stones was 12.01 ± 0.54 for calcium-based, 11.10 ± 0.68 for struvite, and 10.23 ± 0.75 for cystine stones (p < 0.001). Zeff had the best efficacy to separate different classes of stones. The calculated AUC was 0.947 for Zeff; 0.833 for HU40; 0.880 for HU70; and 0.893 for HU140. Zeff derived from rsDECT has superior performance to HU and attenuation ratios for separation of different classes of non-uric-acid stones.

Exponentially Ramsey Sets

We study chromatic numbers of spaces $$\mathbb{R}_p^n=(\mathbb{R}^n, \ell_p)$$ with forbidden monochromatic sets. For some sets, we for the first time obtain explicit exponentially growing lower bounds for the corresponding chromatic numbers; for some others, we substantially improve previously known bounds.

Strong failures of higher analogs of Hindman’s theorem

We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every $X\subseteq\mathbb R$ with $|X|=|\mathbb R|$, and every colour $\gamma\in\mathbb Q$, there are two distinct elements $x_0,x_1$ of $X$ for which $c(x_0+x_1)=\gamma$. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2: For every Abelian group $G$, there exists a colouring $c:G\rightarrow\mathbb Q$ such that for every uncountable $X\subseteq G$, and every colour $\gamma$, for some large enough integer $n$, there are pairwise distinct elements $x_0,\ldots,x_n$ of $X$ such that $c(x_0+\cdots+x_n)=\gamma$. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from $\mathbb Q$ to $\mathbb R$. Theorem 3: Let $\circledast_\kappa$ assert that for every Abelian group $G$ of cardinality $\kappa$, there exists a colouring $c:G\rightarrow G$ such that for every positive integer $n$, every $X_0,\ldots,X_n \in[G]^\kappa$, and every $\gamma\in G$, there are $x_0\in X_0,\ldots, x_n\in X_n$ such that $c(x_0+\cdots+x_n)=\gamma$. Then $\circledast_\kappa$ holds for unboundedly many uncountable cardinals $\kappa$, and it is consistent that $\circledast_\kappa$ holds for all regular uncountable cardinals $\kappa$.

Use of computed tomography gemstone spectral curve in evaluation on histodifferentiation of gastric cancer

To evaluate the application value of spectral curve in gemstone spectral imaging (GSI) in evaluating the differentiation state of gastric cancer. Triple-phase enhanced GSI CT scanning was performed on 65 gastric cancer patients whose differentiation state was later confirmed through gastroscopy. The 101 monochromatic image sets obtained with spectral CT imaging were analyzed with the GSI Viewer software. CT values at 40, 70 and 140 keV images were measured to calculate the slopes for spectral curve in the 40-140, 40-70 and 70-140 keV energy range. They were divided into 3 groups based on the pathological results and the corresponding slope measurements were compared with independent t test. Receiver operating characteristic curves were generated to determine the parameters for optimizing sensitivity and specificity in differentiation state of gastric cancer. As verified pathologically, there were 21 well-differentiated, 22 moderately-differentiated and 22 poorly-differentiated adenocarcinoma cases. The slope for spectral curve was larger at the foci of gastric cancer than that in normal gastric wall: the lower differentiation state, the larger the slope. During the arterial phase, the slope of the normal gastric wall , well-differentiated adenocarcinoma , moderately-differentiated adenocarcinoma and poorly-differentiated adenocarcinoma in the 40-70 keV energy range was 1.78 ± 0.12, 1.86 ± 0.30, 1.87 ± 0.28, 2.59 ± 0.31, the slopes of well and moderately-differentiated adenocarcinoma and normal gastric wall were significantly different from poorly-differentiated adenocarcinoma in the 40-70 keV energy range (P < 0.05), but were not different in the other 2 energy ranges (P > 0.05); during the portal vein phase, the slope of the normal gastric wall, well-differentiated adenocarcinoma, moderately-differentiated adenocarcinoma and poorly-differentiated adenocarcinoma in the 40-70 keV energy range was 3.30 ± 1.13, 3.96 ± 0.29, 4.10 ± 1.28, 4.98 ± 2.15, the slopes in all 3 energy ranges were significantly different in each group (P < 0.05). Using the slopes in the 40-70 keV energy range, the sensitivities and specificities for determining the differentiation states were >81% and >86% respectively during the arterial phase and >72% and >72% during the portal venous phase. The spectral curve at the foci of gastric cancer was correlated with the histological differentiation state. And the slope of spectral curve of GSI may offer a new index for evaluating the differentiation state of gastric cancer before operation.

The Gap Structure of a Family of Integer Subsets

In this paper we investigate the gap structure of a certain family of subsets of $\mathbb{N}$ which produces counterexamples both to the "density version" and the "canonical version" of Brown's lemma. This family includes the members of all complementing pairs of $\mathbb{N}$. We will also relate the asymptotical gap structure of subsets of $\mathbb{N}$ with their density and investigate the asymptotical gap structure of monochromatic and rainbow sets with respect to arbitrary infinite colorings of $\mathbb{N}$.

Residual aneurysm after metal coils treatment detected by spectral CT.

Digital subtraction angiography (DSA) is currently the gold standard for diagnosing the residue or recurrence of aneurysm after treatment, especially in the presence of metal coils. However, DSA is an invasive procedure which may cause additional trauma and economic burden to patients. Spectral CT imaging, as a newly introduced CT imaging mode, produces monochromatic image sets that is able to reduce beam-hardening and other metal-related artifacts, and has found its use in several clinical applications including brain imaging to reduce beam-hardening artifacts. In this study, we describe a case of spectral CT imaging in follow-up of the metal coils treatment and detection of a small leaf of residual aneurysm after metal coils treatment.

High-Definition CT Gemstone Spectral Imaging of the Brain

The purpose of this study was to select the optimal monochromatic level for gemstone spectral imaging (GSI) to minimize both the image noise and beam-hardening artifacts (BHAs) in nonenhanced cranial computed tomography (CT). One hundred subjects scanned with GSI mode on Discovery CT750HD were enrolled. Six sets of CT images were obtained from a single GSI acquisition: conventional 140 kilovolt (peak) (kV[p]) polychromatic images and 5 sets of monochromatic images (80, 75, 70, 65, and 60 kiloelectron volts [keV]). The background noise in the corona and the BHA in 4 different anatomic regions (medulla oblongata, cerebellar, pons, and the inferior part of frontal lobes) were measured and compared between the polychromatic and monochromatic images. Beam-hardening artifact is defined as the square root of the squared noise difference between the region of interest and background. The background noise with the monochromatic sets reduced by -36%, -11%, 11%, 10%, and -14%, respectively, compared to the polychromatic image. For BHA, the reductions were 73%, 91%, 92%, and 80%; 53%, 75%, 65%, and 41.%; 27%, 59%, 44%, and 26%; 7%, 46%, 25%, and 20%; and -14%, 33%, 6%, and 19% for the 4 regions of interest. Both the 70 and 65 keV sets had positive background noise reduction (P = 0.285) and BHA reduction with 70 keV were statistically higher (P < 0.05). In the nonenhanced cranial CT with GSI, the optimal monochromatic level was determined at 70 keV to provide both image noise reduction (11%) and BHA reduction compared to the conventional polychromatic images.

Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v / ∈ N there is a monochromatic path from v to N . We denote by A(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.

Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v / ∈ N there is a monochromatic path from v to N . We denote by A(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.

Monochromatic paths and monochromatic sets of arcs in bipartite tournaments

Monochromatic paths and monochromatic sets of arcs in bipartite tournaments

On four color monochromatic sets with nondecreasing diameter

Let m and r be positive integers. Define f ( m , r ) to be the least positive integer N such that for every coloring of the integers 1 , … , N with r colors there exist monochromatic subsets B 1 and B 2 (not necessarily of the same color), each having m elements, such that (a) max ( B 1 ) - min ( B 1 ) ⩽ max ( B 2 ) - min ( B 2 ) , and (b) max ( B 1 ) < min ( B 2 ) . We improve previous upper bounds to determine that f ( m , 4 ) = 12 m - 9 .

Axiom of choice and chromatic number of the plane

Define a graph U on the set of all points of the plane R as its vertex set, with two points adjacent iff they are distance 1 apart. The graph U ought to be called unit distance plane, and its chromatic number w is called chromatic number of the plane. Finite subgraphs of U are called finite unit distance plane graphs. In 1950 the 18-year old Edward Nelson posed the problem of finding w (see the problem’s history in [Soi1]). A number of relevant results were obtained under additional restrictions on monochromatic sets (see surveys in [CFG,KW,Soi2,Soi3]). Falconer, for example, showed [F] that w is at least 5 if monochromatic sets are Lebesgue measurable. Amazingly though, the problem has withstood all assaults in the general case, leaving us with an embarrassingly wide range for w being 4, 5, 6 or 7. In their fundamental 1951 paper [EB], Erdos and de Bruijn have shown that the chromatic number of the plane is attained on some finite subgraph. This result has naturally channeled much of research in the direction of finite unit distance graphs. One limitation of the Erdos–de-Bruijn result, however, has remained a low key: they ARTICLE IN PRESS

Monochromatic and zero-sum sets of nondecreasing diameter

For positive integers m and r define f(m,r) to be the minimum integer n such that for every coloring of 1, 2, , n with r colors, there exist two monochromatic subsets B1, B2 1, 2, , n (but not nece...